Which of the following statements are true?
(A) The function \(f(x) = \displaystyle\frac{x^4}{4} - \frac{4}{3}x^3 + \frac{x^2}{2} + 6x\) has 3 critical points.
(B) The function \(f(x) = |x| + 3\) has no minimum value.
(C) A local maximum value is always the absolute maximum value.
(D) \(f(x) = x^2\) has minima at \(x = 0\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (D) only
The critical points of the function in (A) are the zeros of its derivative:
\(f'(x) = x^3 - 4x^2 + x + 6\)
Putting \(x = -1\) gives \(-1 - 4 - 1 + 6 = 0\), so \(x + 1\) is a factor:
\(f'(x) = (x + 1)(x^2 - 5x + 6)\)
\(f'(x) = (x + 1)(x - 2)(x - 3)\)
For (B) and (D):
\(|x| + 3 \ge 3 = f(0)\)
\(x^2 \ge 0 = f(0)\)
So the function in (A) has the critical points −1, 2 and 3; \(|x| + 3\) has minimum value 3; and \(x^2\) has its minimum at \(x = 0\).
Checking the statements
(A) The function \(f(x) = \frac{x^4}{4} - \frac{4}{3}x^3 + \frac{x^2}{2} + 6x\) has 3 critical points. — Correct. They are \(x = -1, 2, 3\).
(B) The function \(f(x) = |x| + 3\) has no minimum value. — Incorrect. Its minimum value is 3, at \(x = 0\).
(C) A local maximum value is always the absolute maximum value. — Incorrect. \(x^3 - 3x\) has a local maximum 2 at \(x = -1\), but its value at \(x = 3\) is 18.
(D) \(f(x) = x^2\) has minima at \(x = 0\). — Correct. \(x^2 \ge 0 = f(0)\) for all x.