AP Calculus AB Flashcards: Concavity Of Functions Over Their Domains

Study Concavity Of Functions Over Their Domains in AP Calculus AB with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

AP Calculus AB

Concavity Of Functions Over Their Domains

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QUESTION

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Determine concavity of $f(x) = \text{arcsin}(x)$ at $x = 0$ .

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ANSWER

$f''(x) = \frac{x}{(1-x^2)^{\frac{3}{2}}}$ , $f''(0) = 0$ . Undetermined. Zero second derivative makes concavity indeterminate.

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What this deck covers

This deck focuses on Concavity Of Functions Over Their Domains, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus AB.

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Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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Flashcard 1: Determine concavity of $f(x) = \text{arcsin}(x)$ at $x = 0$ .

Answer: $f''(x) = \frac{x}{(1-x^2)^{\frac{3}{2}}}$ , $f''(0) = 0$ . Undetermined. Zero second derivative makes concavity indeterminate.

Flashcard 2: Determine concavity of $f(x) = e^{-x}$ over its domain.

Answer: $f''(x) = e^{-x} > 0$ , so $f(x)$ is concave up everywhere. Exponential decay function maintains positive concavity.

Flashcard 3: What does it mean for a function to be concave up?

Answer: The graph is curved upwards, like a cup, and $f''(x) > 0$ . Positive second derivative indicates upward curvature.

Flashcard 4: Identify an inflection point for $f(x) = x^3 - 3x^2 + x$ .

Answer: Inflection at $x = 1$ where $f''(x)$ changes sign. Second derivative changes sign at this point.

Flashcard 5: Determine concavity of $f(x) = x^3 - x$ at $x = 1$ .

Answer: $f''(x) = 6x$ , $f''(1) = 6 > 0$ , concave up. Positive second derivative indicates upward concavity.

Flashcard 6: What is the concavity of $f(x) = x^4$ at $x = 1$ ?

Answer: $f''(x) = 12x^2$ , $f''(1) = 12 > 0$ , concave up. Positive second derivative indicates upward concavity.

Flashcard 7: What does it mean for a function to be concave down?

Answer: The graph is curved downwards, like a frown, and $f''(x) < 0$ . Negative second derivative indicates downward curvature.

Flashcard 8: What is the concavity of $f(x) = \text{ln}(x)$ over $x > 0$ ?

Answer: $f''(x) = -\frac{1}{x^2} < 0$ , concave down for $x > 0$ . Natural log is concave down on its domain.

Flashcard 9: Determine concavity of $f(x) = \text{arcsin}(x)$ at $x = 0$ .

Answer: $f''(x) = \frac{x}{(1-x^2)^{\frac{3}{2}}}$ , $f''(0) = 0$ . Undetermined. Zero second derivative makes concavity indeterminate.

Flashcard 10: Determine concavity of $f(x) = \frac{1}{x}$ at $x = 1$ .

Answer: $f''(x) = \frac{2}{x^3}$ , $f''(1) = 2 > 0$ , concave up. Positive second derivative at a point indicates concave up.

Flashcard 11: Identify concavity of $f(x) = \text{tan}(x)$ at $x = 0$ .

Answer: $f''(x) = 2\text{tan}(x)\text{sec}^2(x)$ , $f''(0) = 0$ . Undetermined. Zero second derivative requires further analysis.

Flashcard 12: What is the concavity of $f(x) = -x^2$ over its domain?

Answer: $f''(x) = -2 < 0$ , so $f(x)$ is concave down everywhere. Constant negative second derivative everywhere.

Flashcard 13: Identify concavity of $f(x) = \frac{1}{x^2}$ at $x = 1$ .

Answer: $f''(x) = \frac{6}{x^4}$ , $f''(1) = 6 > 0$ , concave up. Positive second derivative confirms upward concavity.

Flashcard 14: What is the concavity of $f(x) = x^5$ at $x = 0$ ?

Answer: $f''(x) = 20x^3$ , $f''(0) = 0$ . Concavity is undetermined. Zero second derivative makes concavity indeterminate.

Flashcard 15: Determine concavity of $f(x) = e^{-x}$ over its domain.

Answer: $f''(x) = e^{-x} > 0$ , so $f(x)$ is concave up everywhere. Exponential decay function maintains positive concavity.

Flashcard 16: Identify the concavity for $f(x) = \frac{1}{x}$ at $x = -1$ .

Answer: $f''(x) = \frac{2}{x^3}$ , $f''(-1) = -2 < 0$ , concave down. Negative second derivative indicates downward concavity.

Flashcard 17: What is the concavity of $f(x) = 4x^4 - x^2$ at $x = 0$ ?

Answer: $f''(x) = 48x^2 - 2$ , $f''(0) = -2 < 0$ , concave down. Negative second derivative indicates downward concavity.

Flashcard 18: Identify concavity of $f(x) = \text{sin}(x)$ at $x = \text{π}$ .

Answer: $f''(x) = -\text{sin}(x)$ , $f''(\text{π}) = 0$ , undetermined. Zero second derivative makes concavity indeterminate.

Flashcard 19: What is the concavity of $f(x) = x^5$ at $x = 0$ ?

Answer: $f''(x) = 20x^3$ , $f''(0) = 0$ . Concavity is undetermined. Zero second derivative makes concavity indeterminate.

Flashcard 20: Find the concavity of $f(x) = \text{ln}(x)$ at $x = 2$ .

Answer: $f''(x) = -\frac{1}{x^2}$ , $f''(2) < 0$ , concave down. Natural logarithm has negative second derivative.

Flashcard 21: What is the concavity of $f(x) = x^2$ over its domain?

Answer: $f''(x) = 2 > 0$ , so $f(x)$ is concave up everywhere. The second derivative is constant and positive.

Flashcard 22: Identify an inflection point for $f(x) = x^3 - 3x^2 + x$ .

Answer: Inflection at $x = 1$ where $f''(x)$ changes sign. Second derivative changes sign at this point.

Flashcard 23: Determine concavity of $f(x) = x^4 - 4x^2$ at $x = 1$ .

Answer: $f''(x) = 12x^2 - 8$ , $f''(1) = 4 > 0$ , concave up. Positive second derivative at the specified point.

Flashcard 24: What is the inflection point condition for $f(x)$ ?

Answer: $f''(x)$ changes sign at an inflection point. Sign change in $f''(x)$ indicates a change in concavity.

Flashcard 25: What is the concavity of $f(x) = x^3 + 3x^2$ at $x = -1$ ?

Answer: $f''(x) = 6x + 6$ , $f''(-1) = 0$ . Concavity is undetermined. Zero second derivative makes concavity indeterminate.

Flashcard 26: Determine concavity of $f(x) = \frac{1}{x}$ at $x = 1$ .

Answer: $f''(x) = \frac{2}{x^3}$ , $f''(1) = 2 > 0$ , concave up. Positive second derivative at a point indicates concave up.

Flashcard 27: How does the sign of $f''(x)$ affect the graph of $f(x)$ ?

Answer: If $f''(x) > 0$ , $f(x)$ is concave up; if $f''(x) < 0$ , $f(x)$ is concave down. Sign of the second derivative directly determines concavity.

Flashcard 28: What is the concavity of $f(x) = -x^3$ at $x = 1$ ?

Answer: $f''(x) = -6x$ , $f''(1) = -6 < 0$ , concave down. Negative second derivative indicates downward concavity.

Flashcard 29: What is the second derivative test for concavity?

Answer: If $f''(x) > 0$ , $f(x)$ is concave up; if $f''(x) < 0$ , $f(x)$ is concave down. The second derivative determines the direction of curvature.

Flashcard 30: Identify the concavity for $f(x) = \text{cos}(x)$ at $x = \frac{\text{π}}{2}$ .

Answer: $f''(x) = -\text{cos}(x)$ , $f''(\frac{\text{π}}{2}) = 0$ , undetermined. When $f''(x) = 0$ , concavity cannot be determined.

Flashcard 31: Identify concavity of $f(x) = \text{arctan}(x)$ at $x = 0$ .

Answer: $f''(x) = -\frac{2x}{(1+x^2)^2}$ , $f''(0) = 0$ . Undetermined. Zero second derivative makes concavity indeterminate.

Flashcard 32: Identify concavity of $f(x) = \text{sin}(x)$ at $x = 0$ .

Answer: $f''(x) = -\text{sin}(x)$ , $f''(0) = 0$ . Concavity is undetermined. Zero second derivative makes concavity indeterminate.

Flashcard 33: Identify the concavity for $f(x) = x^3$ at $x = 0$ .

Answer: $f''(x) = 6x$ , so $f''(0) = 0$ . Concavity is undetermined. When $f''(x) = 0$ , the second derivative test is inconclusive.

Flashcard 34: Identify the concavity for $f(x) = \frac{1}{x}$ at $x = -1$ .

Answer: $f''(x) = \frac{2}{x^3}$ , $f''(-1) = -2 < 0$ , concave down. Negative second derivative indicates downward concavity.

Flashcard 35: Find the concavity of $f(x) = \text{ln}(x)$ at $x = 2$ .

Answer: $f''(x) = -\frac{1}{x^2}$ , $f''(2) < 0$ , concave down. Natural logarithm has negative second derivative.

Flashcard 36: Identify concavity of $f(x) = \frac{1}{x^2}$ at $x = 1$ .

Answer: $f''(x) = \frac{6}{x^4}$ , $f''(1) = 6 > 0$ , concave up. Positive second derivative confirms upward concavity.

Flashcard 37: Identify concavity of $f(x) = \text{arctan}(x)$ at $x = 0$ .

Answer: $f''(x) = -\frac{2x}{(1+x^2)^2}$ , $f''(0) = 0$ . Undetermined. Zero second derivative makes concavity indeterminate.

Flashcard 38: What is the concavity of $f(x) = x^3 + 3x^2$ at $x = -1$ ?

Answer: $f''(x) = 6x + 6$ , $f''(-1) = 0$ . Concavity is undetermined. Zero second derivative makes concavity indeterminate.

Flashcard 39: Identify concavity of $f(x) = x^2 + 4$ at $x = 0$ .

Answer: $f''(x) = 2 > 0$ , concave up. Constant positive second derivative for all quadratics.

Flashcard 40: Determine concavity of $f(x) = x^2 - 4x$ at $x = 3$ .

Answer: $f''(x) = 2 > 0$ , concave up. Constant positive second derivative for quadratic functions.

Flashcard 41: What is the concavity of $f(x) = -x^2$ over its domain?

Answer: $f''(x) = -2 < 0$ , so $f(x)$ is concave down everywhere. Constant negative second derivative everywhere.

Flashcard 42: Determine concavity of $f(x) = x^3 + 6x^2 + 9x$ at $x = -2$ .

Answer: $f''(x) = 6x + 12$ , $f''(-2) = 0$ . Concavity is undetermined. Zero second derivative makes concavity indeterminate.

Flashcard 43: What is the concavity of $f(x) = -x^3$ at $x = 1$ ?

Answer: $f''(x) = -6x$ , $f''(1) = -6 < 0$ , concave down. Negative second derivative indicates downward concavity.

Flashcard 44: Determine concavity of $f(x) = x^2 - 4x$ at $x = 3$ .

Answer: $f''(x) = 2 > 0$ , concave up. Constant positive second derivative for quadratic functions.

Flashcard 45: Identify concavity of $f(x) = \sin(x)$ at $x = \pi$ .

Answer: $f''(x) = -\sin(x)$ , $f''(\pi) = 0$ , undetermined. Zero second derivative makes concavity indeterminate.

Flashcard 46: Determine concavity of $f(x) = x^3 - x$ at $x = 1$ .

Answer: $f''(x) = 6x$ , $f''(1) = 6 > 0$ , concave up. Positive second derivative indicates upward concavity.

Flashcard 47: Identify the concavity for $f(x) = x^3$ at $x = 0$ .

Answer: $f''(x) = 6x$ , so $f''(0) = 0$ . Concavity is undetermined. When $f''(x) = 0$ , the second derivative test is inconclusive.

Flashcard 48: What is the concavity of $f(x) = x^2$ over its domain?

Answer: $f''(x) = 2 > 0$ , so $f(x)$ is concave up everywhere. The second derivative is constant and positive.

Flashcard 49: At what point does concavity change for $f(x) = x^3$ ?

Answer: Concavity changes at $x = 0$ , where $f''(x) = 0$ . Inflection occurs where $f''(x) = 0$ and changes sign.

Flashcard 50: Identify concavity of $f(x) = \text{tan}(x)$ at $x = 0$ .

Answer: $f''(x) = 2\text{tan}(x)\text{sec}^2(x)$ , $f''(0) = 0$ . Undetermined. Zero second derivative requires further analysis.

Flashcard 51: Determine concavity of $f(x) = e^x$ over its domain.

Answer: $f''(x) = e^x > 0$ , so $f(x)$ is concave up everywhere. The exponential function has constant positive concavity.

Flashcard 52: Identify concavity of $f(x) = x^3 - 3x$ at $x = -1$ .

Answer: $f''(x) = 6x$ , $f''(-1) = -6 < 0$ , concave down. Negative second derivative indicates downward concavity.

Flashcard 53: Determine concavity of $f(x) = x^3 + 6x^2 + 9x$ at $x = -2$ .

Answer: $f''(x) = 6x + 12$ , $f''(-2) = 0$ . Concavity is undetermined. Zero second derivative makes concavity indeterminate.

Flashcard 54: Determine concavity of $f(x) = x^4 - 4x^2$ at $x = 1$ .

Answer: $f''(x) = 12x^2 - 8$ , $f''(1) = 4 > 0$ , concave up. Positive second derivative at the specified point.

Flashcard 55: What is the concavity of $f(x) = 4x^4 - x^2$ at $x = 0$ ?

Answer: $f''(x) = 48x^2 - 2$ , $f''(0) = -2 < 0$ , concave down. Negative second derivative indicates downward concavity.

Flashcard 56: Identify concavity of $f(x) = x^3 - 3x$ at $x = -1$ .

Answer: $f''(x) = 6x$ , $f''(-1) = -6 < 0$ , concave down. Negative second derivative indicates downward concavity.

Flashcard 57: Determine concavity of $f(x) = e^x$ over its domain.

Answer: $f''(x) = e^x > 0$ , so $f(x)$ is concave up everywhere. The exponential function has constant positive concavity.

Flashcard 58: Identify the concavity for $f(x) = \text{cos}(x)$ at $x = \frac{\text{π}}{2}$ .

Answer: $f''(x) = -\text{cos}(x)$ , $f''(\frac{\text{π}}{2}) = 0$ , undetermined. When $f''(x) = 0$ , concavity cannot be determined.

Flashcard 59: What does it mean for a function to be concave up?

Answer: The graph is curved upwards, like a cup, and $f''(x) > 0$ . Positive second derivative indicates upward curvature.

Flashcard 60: Identify concavity of $f(x) = x^2 + 4$ at $x = 0$ .

Answer: $f''(x) = 2 > 0$ , concave up. Constant positive second derivative for all quadratics.

Flashcard 61: What is the inflection point condition for $f(x)$ ?

Answer: $f''(x)$ changes sign at an inflection point. Sign change in $f''(x)$ indicates a change in concavity.

Flashcard 62: What is the concavity of $f(x) = x^4$ at $x = 1$ ?

Answer: $f''(x) = 12x^2$ , $f''(1) = 12 > 0$ , concave up. Positive second derivative indicates upward concavity.

Flashcard 63: What does it mean for a function to be concave down?

Answer: The graph is curved downwards, like a frown, and $f''(x) < 0$ . Negative second derivative indicates downward curvature.

Flashcard 64: How does the sign of $f''(x)$ affect the graph of $f(x)$ ?

Answer: If $f''(x) > 0$ , $f(x)$ is concave up; if $f''(x) < 0$ , $f(x)$ is concave down. Sign of the second derivative directly determines concavity.

Flashcard 65: What is the concavity of $f(x) = \text{ln}(x)$ over $x > 0$ ?

Answer: $f''(x) = -\frac{1}{x^2} < 0$ , concave down for $x > 0$ . Natural log is concave down on its domain.

Flashcard 66: What is the second derivative test for concavity?

Answer: If $f''(x) > 0$ , $f(x)$ is concave up; if $f''(x) < 0$ , $f(x)$ is concave down. The second derivative determines the direction of curvature.

Flashcard 67: Identify concavity of $f(x) = \text{sin}(x)$ at $x = 0$ .

Answer: $f''(x) = -\text{sin}(x)$ , $f''(0) = 0$ . Concavity is undetermined. Zero second derivative makes concavity indeterminate.

Flashcard 68: At what point does concavity change for $f(x) = x^3$ ?

Answer: Concavity changes at $x = 0$ , where $f''(x) = 0$ . Inflection occurs where $f''(x) = 0$ and changes sign.

AP Calculus AB Flashcards: Concavity Of Functions Over Their Domains

Concavity Of Functions Over Their Domains

What this deck covers

How to use these flashcards

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Flashcard 1: Determine concavity of f(x)=arcsin(x)f(x) = \text{arcsin}(x)f(x)=arcsin(x) at x=0x = 0x=0.

Flashcard 2: Determine concavity of f(x)=e−xf(x) = e^{-x}f(x)=e−x over its domain.

Flashcard 3: What does it mean for a function to be concave up?

Flashcard 4: Identify an inflection point for f(x)=x3−3x2+xf(x) = x^3 - 3x^2 + xf(x)=x3−3x2+x.

Flashcard 5: Determine concavity of f(x)=x3−xf(x) = x^3 - xf(x)=x3−x at x=1x = 1x=1.

Flashcard 6: What is the concavity of f(x)=x4f(x) = x^4f(x)=x4 at x=1x = 1x=1?

Flashcard 7: What does it mean for a function to be concave down?

Flashcard 8: What is the concavity of f(x)=ln(x)f(x) = \text{ln}(x)f(x)=ln(x) over x>0x > 0x>0?

Flashcard 9: Determine concavity of f(x)=arcsin(x)f(x) = \text{arcsin}(x)f(x)=arcsin(x) at x=0x = 0x=0.

Flashcard 10: Determine concavity of f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ at x=1x = 1x=1.

Flashcard 11: Identify concavity of f(x)=tan(x)f(x) = \text{tan}(x)f(x)=tan(x) at x=0x = 0x=0.

Flashcard 12: What is the concavity of f(x)=−x2f(x) = -x^2f(x)=−x2 over its domain?

Flashcard 13: Identify concavity of f(x)=1x2f(x) = \frac{1}{x^2}f(x)=x21​ at x=1x = 1x=1.

Flashcard 14: What is the concavity of f(x)=x5f(x) = x^5f(x)=x5 at x=0x = 0x=0?

Flashcard 15: Determine concavity of f(x)=e−xf(x) = e^{-x}f(x)=e−x over its domain.

Flashcard 16: Identify the concavity for f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ at x=−1x = -1x=−1.

Flashcard 17: What is the concavity of f(x)=4x4−x2f(x) = 4x^4 - x^2f(x)=4x4−x2 at x=0x = 0x=0?

Flashcard 18: Identify concavity of f(x)=sin(x)f(x) = \text{sin}(x)f(x)=sin(x) at x=πx = \text{π}x=π.

Flashcard 19: What is the concavity of f(x)=x5f(x) = x^5f(x)=x5 at x=0x = 0x=0?

Flashcard 20: Find the concavity of f(x)=ln(x)f(x) = \text{ln}(x)f(x)=ln(x) at x=2x = 2x=2.

Flashcard 21: What is the concavity of f(x)=x2f(x) = x^2f(x)=x2 over its domain?

Flashcard 22: Identify an inflection point for f(x)=x3−3x2+xf(x) = x^3 - 3x^2 + xf(x)=x3−3x2+x.

Flashcard 23: Determine concavity of f(x)=x4−4x2f(x) = x^4 - 4x^2f(x)=x4−4x2 at x=1x = 1x=1.

Flashcard 24: What is the inflection point condition for f(x)f(x)f(x)?

Flashcard 25: What is the concavity of f(x)=x3+3x2f(x) = x^3 + 3x^2f(x)=x3+3x2 at x=−1x = -1x=−1?

Flashcard 26: Determine concavity of f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ at x=1x = 1x=1.

Flashcard 27: How does the sign of f′′(x)f''(x)f′′(x) affect the graph of f(x)f(x)f(x)?

Flashcard 28: What is the concavity of f(x)=−x3f(x) = -x^3f(x)=−x3 at x=1x = 1x=1?

Flashcard 29: What is the second derivative test for concavity?

Flashcard 30: Identify the concavity for f(x)=cos(x)f(x) = \text{cos}(x)f(x)=cos(x) at x=π2x = \frac{\text{π}}{2}x=2π​.

Flashcard 31: Identify concavity of f(x)=arctan(x)f(x) = \text{arctan}(x)f(x)=arctan(x) at x=0x = 0x=0.

Flashcard 32: Identify concavity of f(x)=sin(x)f(x) = \text{sin}(x)f(x)=sin(x) at x=0x = 0x=0.

Flashcard 33: Identify the concavity for f(x)=x3f(x) = x^3f(x)=x3 at x=0x = 0x=0.

Flashcard 34: Identify the concavity for f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ at x=−1x = -1x=−1.

Flashcard 35: Find the concavity of f(x)=ln(x)f(x) = \text{ln}(x)f(x)=ln(x) at x=2x = 2x=2.

Flashcard 36: Identify concavity of f(x)=1x2f(x) = \frac{1}{x^2}f(x)=x21​ at x=1x = 1x=1.

Flashcard 37: Identify concavity of f(x)=arctan(x)f(x) = \text{arctan}(x)f(x)=arctan(x) at x=0x = 0x=0.

Flashcard 38: What is the concavity of f(x)=x3+3x2f(x) = x^3 + 3x^2f(x)=x3+3x2 at x=−1x = -1x=−1?

Flashcard 39: Identify concavity of f(x)=x2+4f(x) = x^2 + 4f(x)=x2+4 at x=0x = 0x=0.

Flashcard 40: Determine concavity of f(x)=x2−4xf(x) = x^2 - 4xf(x)=x2−4x at x=3x = 3x=3.

Flashcard 41: What is the concavity of f(x)=−x2f(x) = -x^2f(x)=−x2 over its domain?

Flashcard 42: Determine concavity of f(x)=x3+6x2+9xf(x) = x^3 + 6x^2 + 9xf(x)=x3+6x2+9x at x=−2x = -2x=−2.

Flashcard 43: What is the concavity of f(x)=−x3f(x) = -x^3f(x)=−x3 at x=1x = 1x=1?

Flashcard 44: Determine concavity of f(x)=x2−4xf(x) = x^2 - 4xf(x)=x2−4x at x=3x = 3x=3.

Flashcard 45: Identify concavity of f(x)=sin⁡(x)f(x) = \sin(x)f(x)=sin(x) at x=πx = \pix=π.

Flashcard 46: Determine concavity of f(x)=x3−xf(x) = x^3 - xf(x)=x3−x at x=1x = 1x=1.

Flashcard 47: Identify the concavity for f(x)=x3f(x) = x^3f(x)=x3 at x=0x = 0x=0.

Flashcard 48: What is the concavity of f(x)=x2f(x) = x^2f(x)=x2 over its domain?

Flashcard 49: At what point does concavity change for f(x)=x3f(x) = x^3f(x)=x3?

Flashcard 50: Identify concavity of f(x)=tan(x)f(x) = \text{tan}(x)f(x)=tan(x) at x=0x = 0x=0.

Flashcard 51: Determine concavity of f(x)=exf(x) = e^xf(x)=ex over its domain.

Flashcard 52: Identify concavity of f(x)=x3−3xf(x) = x^3 - 3xf(x)=x3−3x at x=−1x = -1x=−1.

Flashcard 53: Determine concavity of f(x)=x3+6x2+9xf(x) = x^3 + 6x^2 + 9xf(x)=x3+6x2+9x at x=−2x = -2x=−2.

Flashcard 54: Determine concavity of f(x)=x4−4x2f(x) = x^4 - 4x^2f(x)=x4−4x2 at x=1x = 1x=1.

Flashcard 55: What is the concavity of f(x)=4x4−x2f(x) = 4x^4 - x^2f(x)=4x4−x2 at x=0x = 0x=0?

Flashcard 56: Identify concavity of f(x)=x3−3xf(x) = x^3 - 3xf(x)=x3−3x at x=−1x = -1x=−1.

Flashcard 57: Determine concavity of f(x)=exf(x) = e^xf(x)=ex over its domain.

Flashcard 58: Identify the concavity for f(x)=cos(x)f(x) = \text{cos}(x)f(x)=cos(x) at x=π2x = \frac{\text{π}}{2}x=2π​.