AP Calculus AB: Increasing/Decreasing and First Derivative Test — Drill 1

About This Drill

AP Calculus AB: Increasing/Decreasing and First Derivative Test — Drill 1 is a Math practice drill covering First Derivative Test. It contains 5 original questions created by Brian Stewart, a Barron's test prep author with over 20 years of tutoring experience.

Practice using the first derivative to identify intervals where a function is increasing or decreasing and to classify critical points as relative maxima, minima, or neither. These AP Calculus AB skills appear frequently on both the multiple-choice and free-response sections.

Questions in This Drill

1. Let \( f \) be a differentiable function with derivative \( f'(x) = (x-1)(x+3) \). On which of the following intervals is \( f \) increasing?
2. The function \( g \) is continuous on \( (-\infty, \infty) \). The sign of \( g'(x) \) is positive for \( x 2 \). Which of the following must be true?
3. Let \( f(x) = x^3 - 6x^2 + 9x + 1 \). Which of the following correctly identifies the relative extrema of \( f \)?
4. Let \( h(x) = x^3 \). Which of the following correctly describes the behavior of \( h \) at \( x = 0 \)?
5. The function \( f \) is differentiable on \( (-\infty, \infty) \). The table below shows the sign of \( f'(x) \) on selected intervals.
   
   

   Interval	(-∞, -2)	(-2, 0)	(0, 5)	(5, ∞)
   Sign of f'(x)	−	−	+	−

   
   Which of the following correctly describes the relative extrema of \( f \)?