Mathematics

Problem

Let

g(x)=\displaystyle\int_{0}^{x} f(t)\,dt

. The function

f

satisfies

f(x)<0

(-3,-1)

and

(2,5)

and

f(x)>0

for all other

x

. Determine the open interval(s) on which

g

is decreasing.

Intuitive explanation

The key insight is that $g$ is the antiderivative of $f$ , so $g'(x) = f(x)$ by the Fundamental Theorem of Calculus. A function decreases exactly where its derivative is negative. Since you know where $f$ is negative, you've already identified where $g$ must be decreasing—no integration needed, just apply the relationship between a function and its derivative.

Intuitive explanation

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