this is the graph of f. let g(x)=∫₀ˣ f(t)dt. what is an appropriate calculus - based justification for the…

this is the graph of f. let g(x)=∫₀ˣ f(t)dt. what is an appropriate calculus - based justification for the fact that g has an inflection point at x = c? choose 1 answer: a f is positive. b f has an inflection point at x = c. c f has a relative minimum at x = c.

this is the graph of f. let g(x)=∫₀ˣ f(t)dt. what is an appropriate calculus - based justification for the fact that g has an inflection point at x = c? choose 1 answer: a f is positive. b f has an inflection point at x = c. c f has a relative minimum at x = c.

Answer

Explanation:

Step1: Recall the fundamental theorem of calculus

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

Step2: Recall the definition of inflection point

An inflection point of $g(x)$ occurs where $g^{\prime\prime}(x)$ changes sign. Since $g^{\prime\prime}(x) = f^\prime(x)$, an inflection point of $g$ occurs where $f^\prime(x)$ changes sign. A relative minimum of $f$ occurs where $f^\prime(x)$ changes sign from negative to positive. When $f$ has a relative minimum at $x = c$, $f^\prime(x)$ changes sign at $x = c$. So $g^{\prime\prime}(x)$ changes sign at $x = c$, and $g$ has an inflection point at $x = c$.

Answer:

C. $f$ has a relative minimum at $x = c$.