The function $f(x) = x^2 - 4x + 6$ is

(A) Strictly decreasing on $(-∞, 2) ∪ (2,∞)$
(B) Strictly increasing on $(2,∞)$
(C) Strictly increasing on $(-∞, ∞)$
(D) Strictly decreasing on $(-∞,2)$

Choose the correct answer from the options given below:

(B) and (D) only

The correct answer is Option (2) → (B) and (D) only

(B) Strictly increasing on $(2,∞)$
(D) Strictly decreasing on $(-∞,2)$

Given function: $f(x) = x^2 - 4x + 6$

Differentiate: $f'(x) = 2x - 4$

Set $f'(x) = 0 \Rightarrow 2x - 4 = 0 \Rightarrow x = 2$

So the derivative changes sign at $x = 2$

Test intervals:

• For $x < 2$, say $x = 1$: $f'(1) = 2(1) - 4 = -2$ ⇒ decreasing
• For $x > 2$, say $x = 3$: $f'(3) = 2(3) - 4 = 2$ ⇒ increasing

Therefore:

• $f(x)$ is strictly decreasing on $(-\infty,\ 2)$
• $f(x)$ is strictly increasing on $(2,\ \infty)$