Let \(f: \mathrm{R} \to \mathrm{R}\) be defined as \(f(x)=4x^2-16x+5\), then
A. Maximum value of f(x) is −11.
B. Minimum value of f(x) is −11.
C. Minimum value of f(x) is 20.
D. No maximum value of f(x).
Choose the correct answer from the options given below
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → B and D only
Differentiating:
\(f'(x) = 8x - 16 = 0\)
\(x = 2\)
\(f''(x) = 8 \gt 0\), so x = 2 is a point of minimum
\(f(2) = 16 - 32 + 5 = -11\)
The graph is an upward parabola, so f(x) grows without limit as x moves away from 2.
So the minimum value of f is −11 (at x = 2), and f has no maximum value.
Checking the statements
(A) Maximum value of f(x) is −11. — Incorrect. −11 is the minimum; f(0) = 5 is already larger.
(B) Minimum value of f(x) is −11. — Correct. f(2) = −11 and \(f''(x) = 8 \gt 0\).
(C) Minimum value of f(x) is 20. — Incorrect. The minimum is −11.
(D) No maximum value of f(x). — Correct. An upward parabola has no highest point.