The twice differentiable function $g$ and its second derivative $g^{\prime \prime}$ are graphed.

Given that $g^{\prime}(c)=0$, what is an appropriate calculus-based justification for the fact that $g$ has a relative maximum at $x=c$ ?

Choose 1 answer:
(A) $g^{\prime \prime}(c)

Q: The twice differentiable function $g$ and its second derivative $g^{\prime \prime}$ are graphed. Given that $g^{\prime}(c)=0$, what is an appropriate calculus-based justification for the fact that $g$ has a relative maximum at $x=c$ ? Choose 1 answer: (A) $g^{\prime \prime}(c)<0$ (B) $g$ is concave down at $x=c$. (C) $g^{\prime \prime}$ is concave up at $x=c$.

∻Solution by Steps∻ ‖ step 1 ⋮ Given that $g'(c) = 0$, we need to determine the nature of the critical point at $x = c$. ‖ ‖ step 2 ⋮ The second derivative test states that if $g''(c) < 0$, then $g$ has a relative maximum at $x = c$. ‖ ‖ step 3 ⋮ From the image, $g''(c)$ is below the x-axis to the right of $c$, indicating $g''(c) < 0$. ‖ ‖ step 4 ⋮ Therefore, $g$ has a relative maximum at $x = c$ because $g''(c)

Question

The twice differentiable function $g$ and its second derivative $g^{\prime \prime}$ are graphed.

Given that $g^{\prime}(c)=0$, what is an appropriate calculus-based justification for the fact that $g$ has a relative maximum at $x=c$ ?

Choose 1 answer:
(A) $g^{\prime \prime}(c)<0$
(B) $g$ is concave down at $x=c$.
(C) $g^{\prime \prime}$ is concave up at $x=c$.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Given that $g'(c) = 0$, we need to determine the nature of the critical point at $x = c$. ‖ ‖ step 2 ⋮ The second derivative test states that if $g''(c) < 0$, then $g$ has a relative maximum at $x = c$. ‖ ‖ step 3 ⋮ From the image, $g''(c)$ is below the x-axis to the right of $c$, indicating $g''(c) < 0$. ‖ ‖ step 4 ⋮ Therefore, $g$ has a relative maximum at $x = c$ because $g''(c) < 0$. ‖ ***A*** ∻Key Concept∻ ⚹Second Derivative Test⚹ ∻Explanation∻ ⚹The second derivative test helps determine the concavity of a function at a critical point. If $g''(c) < 0$, the function is concave down at $c$, indicating a relative maximum.⚹◊What is the mathematical definition of a relative maximum for a function $g$ at $x=c$?⍭ Generate me a similar question◊

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