Mathematics

Mathematics

Problem

For a continuous function ff defined on an interval containing aa, let F(x)=∫axf(t) dtF(x)=\int_{a}^{x} f(t)\,dt. Which of the following statements is always true? a) If FF is decreasing at x=cx=c, then f(c)>0f(c)>0. b) If ff is decreasing on an interval, then FF is decreasing on that interval. c) If f(x)>0f(x)>0 on an interval, then FF is increasing on that interval. d) If f(x)<0f(x)<0 at x=cx=c, then FF has a local minimum at cc.
Intuitive explanation

The Fundamental Theorem of Calculus is the bridge here—it tells you that the derivative of an accumulation function F(x)F(x) recovers the original function f(x)f(x). Rather than thinking about what FF looks like geometrically, focus on what happens when you differentiate it: the key insight is that F′(x)=f(x)F'(x) = f(x) whenever ff is continuous. This relationship holds regardless of what ff does, so any statement claiming FF is differentiable with this derivative will always be true.

Intuitive explanation

The Fundamental Theorem of Calculus is the bridge here—it tells you that the derivative of an accumulation function F(x)F(x) recovers the original function f(x)f(x). Rather than thinking about what FF looks like geometrically, focus on what happens when you differentiate it: the key insight is that F′(x)=f(x)F'(x) = f(x) whenever ff is continuous. This relationship holds regardless of what ff does, so any statement claiming FF is differentiable with this derivative will always be true.

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