For $x ∈R$, if $f(x)=-(x-1)^2+2$, then

(A) $f$ is an increasing function on $(-∞,1]$
(B) $f$ has no critical points
(C) $f$ has a maximum value at $x = 1$
(D) $f$ has a minimum value at $x = 1$

Choose the correct answer from the options given below:

(A) and (C) only

The correct answer is Option (3) → (A) and (C) only

Given: $f(x) = - (x - 1)^2 + 2$

First derivative:

$f'(x) = -2(x - 1)$

Set $f'(x) = 0$ ⟹ $x = 1$ is a critical point.

Sign of $f'(x)$:

• For $x < 1$: $f'(x) > 0$ ⟹ function increasing

• For $x > 1$: $f'(x) < 0$ ⟹ function decreasing

⟹ First derivative test confirms a maximum at $x = 1$.

Second derivative:

$f''(x) = -2$ (always negative)

⟹ Second derivative test confirms maximum at $x = 1$