If \(f(x) = x^2 - 4x + 13,\ x \in R\), then which of the following are correct?
(A) \(x = 2\) is a stationary point of f(x).
(B) f(x) is increasing function on \((2, \infty)\)
(C) f(x) have maxima at \(x = 2\)
(D) \(f(2) = 9\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (B) and (D) only
\( \displaystyle f'(x) = 2x - 4 = 2(x - 2) \)
\( \displaystyle f''(x) = 2 \)
\( \displaystyle f(2) = 4 - 8 + 13 = 9 \)
So \(f'(2) = 0\), \(f'(x) \gt 0\) for \(x \gt 2\), \(f''(2) = 2 \gt 0\) and \(f(2) = 9\).
Checking the statements
(A) \(x = 2\) is a stationary point of f(x). — Correct. \(f'(2) = 0\), so x = 2 is a stationary point.
(B) f(x) is increasing function on \((2, \infty)\) — Correct. For \(x \gt 2\), \(f'(x) = 2(x - 2) \gt 0\).
(C) f(x) have maxima at \(x = 2\) — Incorrect. \(f''(2) = 2 \gt 0\), so x = 2 is a point of local minimum, not maximum.
(D) \(f(2) = 9\) — Correct. \(f(2) = 4 - 8 + 13 = 9\).