AP Calculus AB · Question of the Day

AP Calculus AB Question of the Day

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Monday, September 21, 2026

For $x\ne0$ , $f'(x)=\dfrac{(x-3)(x+2)}{x^2}$ . On which interval(s) is $f$ increasing?

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Question of the Day

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For $x\ne0$ , $f'(x)=\dfrac{(x-3)(x+2)}{x^2}$ . On which interval(s) is $f$ increasing?

$(-\infty,-2)\cup(3,\infty)$ (correct answer)
$(-2,0)\cup(0,3)$
$(-\infty,0)\cup(0,\infty)$
$( -2,3)$
$(-\infty,-2)\cup(-2,0)$

Explanation: This problem tests your ability to determine the intervals where a function is increasing or decreasing by analyzing the sign of its first derivative. A function f is increasing on intervals where f'(x) > 0 and decreasing where f'(x) < 0. For f'(x) = (x-3)(x+2)/x^2, the denominator is positive except at x=0 (undefined), so the sign matches the numerator (x-3)(x+2). Critical points are x=-2, x=0, and x=3, with positivity in (-∞,-2) and (3,∞). Thus, f is increasing on (-∞,-2) ∪ (3,∞). A tempting distractor is choice B (-2,0) ∪ (0,3), but that's where f'(x) < 0, possibly from overlooking the denominator's consistent positivity. Always create a sign chart marking roots and undefined points, then test a point in each interval to determine the derivative's sign systematically.