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

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 ：\(\boxed{\text{The critical point x=7 is a minimum}}\)