Let \(f(x) = x^3 + 15x^2 - 33x - 36 \) be a real-valued function. Which of the…
2023
Let
\(f(x) = x^3 + 15x^2 - 33x - 36
\)
be a real-valued function. Which of the following statements is/are TRUE?
- A.
\(f(x)\)does not have a local maximum. - B.
\(f(x)\)has a local maximum. - C.
\(f(x)\)does not have a local minimum. - D.
\(f(x)\)has a local minimum.
Show answer & explanation
Correct answer: B, D
Key steps: use the derivative and the second-derivative test to classify critical points.
Compute the derivative: f'(x)=3x^2+30x-33 = 3(x^2+10x-11). Solve f'(x)=0 to get x=1 and x=-11.
Second derivative: f''(x)=6x+30. Evaluate at critical points:
f''(1)=36 > 0, so there is a local minimum at x = 1.
f''(-11)=-36 < 0, so there is a local maximum at x = -11.
Conclusion: The statements 'f(x) has a local maximum.' and 'f(x) has a local minimum.' are true; the statements denying the existence of those extrema are false.