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Posted 2 months ago

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

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

Choose 1 answer:
(A) g(c)<0g^{\prime \prime}(c)<0
(B) gg is concave down at x=cx=c.
(C) gg^{\prime \prime} is concave up at x=cx=c.
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Answer from Sia
Posted 2 months ago
Solution by Steps
step 2
The second derivative test states that if g(c)<0g''(c) < 0, then gg has a relative maximum at x=cx = c
step 3
From the image, g(c)g''(c) is below the x-axis to the right of cc, indicating g(c)<0g''(c) < 0
step 4
Therefore, gg has a relative maximum at x=cx = c because g(c)<0g''(c) < 0
Key Concept
Second Derivative Test
The second derivative test helps determine the concavity of a function at a critical point. If g(c)<0g''(c) < 0, the function is concave down at cc, indicating a relative maximum.

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