Question

Math

Posted 6 months ago

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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 $x=c$.
```

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Answer from Sia

Posted 6 months ago

Solution by Steps

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.

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