The function \(f(x) = 5 + 4x - x^{2}\) is strictly increasing in the interval
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \((-\infty, 2)\)
A function is strictly increasing where \(f'(x) \gt 0\).
\( f'(x) = 4 - 2x \)
\( 4 - 2x \gt 0 \iff x \lt 2 \)
So f is strictly increasing in \((-\infty, 2)\) and strictly decreasing in \((2, \infty)\).
Why the other options are incorrect
Option (1) \((2, \infty)\) — Incorrect. For x > 2, f′(x) = 4 − 2x < 0, so f is decreasing there.
Option (2) \((-2, \infty)\) — Incorrect. It contains x = 3, where f′(3) = −2 < 0.
Option (3) \((-\infty, 4)\) — Incorrect. It contains (2, 4), where f is decreasing: f′(3) = −2 < 0.