AP Calculus BC 5.7 Using the Second Derivative Test to Determine Extrema Questions

Review the second derivative test by evaluating f double prime at critical points and classifying local extrema, including cases where the test is inconclusive.

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FUN-4.A—Justify conclusions about the behavior of a function based on the behavior of its derivatives—Topic 5.7

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Consider the differential equation $\frac{d y}{d x}=x^{2}-\frac{1}{2} y$ .

Let y=f(x) be the particular solution to the given differential equation whose graph passes through the point (-2, 8). Does the graph of f have a relative minimum, a relative maximum, or neither at the point (-2, 8) ? Justify your answer.

$\left.\frac{d y}{d x}\right|_{(x, y)=(-2,8)}=(-2)^{2}-\frac{1}{2} \cdot 8=0$

\left.\frac{d^{2} y}{d x^{2}}\right|_{(x, y)=(-2,8)}=2(-2)-\frac{1}{2}\left((-2)^{2}-\frac{1}{2} \cdot 8\right)=-4<0

Thus, the graph of f has a relative maximum at the point (-2, 8).

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