This is the graph of $f$.

Let $g(x)=\int_{0}^{x} f(t) d t$.
What is an appropriate calculus-based justification for the fact that $g$ is concave up?

Choose 1 answer:
(A) $f$ is positive.
(B) $f$ is increasing.
(c) $f$ is concave up.

Question

This is the graph of $f$.

Let $g(x)=\int_{0}^{x} f(t) d t$.
What is an appropriate calculus-based justification for the fact that $g$ is concave up?

Choose 1 answer:
(A) $f$ is positive.
(B) $f$ is increasing.
(c) $f$ is concave up.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ We are given the function $g(x) = \int_{0}^{x} f(t) \, dt$ and need to determine why $g$ is concave up. ‖ ‖ step 2 ⋮ To determine concavity, we need to consider the second derivative of $g(x)$. ‖ ‖ step 3 ⋮ By the Fundamental Theorem of Calculus, the first derivative of $g(x)$ is $g'(x) = f(x)$. ‖ ‖ step 4 ⋮ The second derivative of $g(x)$ is $g''(x) = f'(x)$. ‖ ‖ step 5 ⋮ For $g(x)$ to be concave up, $g''(x) > 0$. This means $f'(x) > 0$, indicating that $f(x)$ is increasing. ‖ ***B*** ∻Key Concept∻ ⚹Concavity and Second Derivative⚹ ∻Explanation∻ ⚹A function $g(x)$ is concave up if its second derivative $g''(x)$ is positive. Since $g''(x) = f'(x)$, $f(x)$ must be increasing for $g(x)$ to be concave up.⚹◊Can you explain how the concavity of the function g(x) relates to the concavity of its derivative f(x⍭ Generate me a similar question◊

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