Let $f(x)=2 x^{3}-36 x^{2}+210 x+84$
(a) The second derivative at $x=7$ is
\[
f^{\prime \prime}(7)=
\]
(b) The critical point $x=7$ is
a maximum
a minimum
unknown because the test is inconclusive

Answer

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Answer

$\boxed{\text{The critical point x=7 is a minimum}}$

Steps

Step 1 ：Find the first and second derivatives of the function f(x):

Step 2 ：$f'(x) = 6x^2 - 72x + 210$

Step 3 ：$f''(x) = 12x - 72$

Step 4 ：Evaluate the second derivative at x=7:

Step 5 ：$f''(7) = 12(7) - 72 = 12$

Step 6 ：$\boxed{f''(7) = 12}$

Step 7 ：Determine if the critical point x=7 is a maximum, minimum, or unknown based on the second derivative test:

Step 8 ：Since $f''(7) > 0$, the critical point x=7 is a minimum.

Step 9 ：$\boxed{\text{The critical point x=7 is a minimum}}$