let g(x)=∫₀ˣ f(t)dt, where the graph of f is shown below. on what interval(s) is g both concave up and…

let g(x)=∫₀ˣ f(t)dt, where the graph of f is shown below. on what interval(s) is g both concave up and increasing? graph of f (-2,0) (-2,0),(5,10) (-2,0),(8,10) cannot be determined

let g(x)=∫₀ˣ f(t)dt, where the graph of f is shown below. on what interval(s) is g both concave up and increasing? graph of f (-2,0) (-2,0),(5,10) (-2,0),(8,10) cannot be determined

Answer

Explanation:

Step1: Recall fundamental theorem of calculus

By the fundamental - theorem of calculus, $g^\prime(x)=f(x)$ and $g^{\prime\prime}(x)=f^\prime(x)$.

Step2: Determine when $g(x)$ is increasing

$g(x)$ is increasing when $g^\prime(x)=f(x)>0$. From the graph of $f(x)$, $f(x)>0$ on the intervals $(- 2,0)$ and $(8,10)$.

Step3: Determine when $g(x)$ is concave - up

$g(x)$ is concave - up when $g^{\prime\prime}(x)=f^\prime(x)>0$. From the graph of $f(x)$, $f(x)$ has a positive slope (i.e., $f^\prime(x)>0$) on the intervals $(-2,0)$ and $(8,10)$.

Answer:

$(-2,0),(8,10)$