Mathematics

Mathematics

Problem

A differentiable function ff satisfies the following: • ff is decreasing on (−∞,−3)(-\infty,-3), increasing on (−3,1)(-3,1), and decreasing on (1,∞)(1,\infty). • ff is concave up on (−∞,0)(-\infty,0) and concave down on (0,∞)(0,\infty). Identify all xx-values where f′(x)=0f'(x)=0, and state whether f′(x)f'(x) is positive or negative on each of the open intervals determined by these zeros.
Intuitive explanation

The key is connecting the monotonicity of ff directly to the sign of its derivative: where ff decreases, f′<0f'<0, and where ff increases, f′>0f'>0. The transition points where ff changes from decreasing to increasing (or vice versa) are exactly where f′=0f'=0. The concavity information tells you about f′′f'', but it doesn't change where f′f' equals zero—it only describes the shape of ff between those critical points.

Intuitive explanation

The key is connecting the monotonicity of ff directly to the sign of its derivative: where ff decreases, f′<0f'<0, and where ff increases, f′>0f'>0. The transition points where ff changes from decreasing to increasing (or vice versa) are exactly where f′=0f'=0. The concavity information tells you about f′′f'', but it doesn't change where f′f' equals zero—it only describes the shape of ff between those critical points.

Browse the problem library