Assertion (A): The function f(x)=x^2-4 x+6 is strictly increasing… - Vidyadip

MCQ

Assertion (A): The function $f(x)=x^2-4 x+6$ is strictly increasing in the interval $(2, \infty)$.
Reason (R): The function $f(x)=x^2-4 x+6$ is strictly decreasing in the interval $(-\infty, 2)$.

A
Both A and R are true and R is the correct explanation of A.
B
Both A and R are true but R is not the correct explanation of A
C
A is true but R is false.
D
A is false but R is true.

✓

Answer

(b) Both A and R are true but R is not the correct explanation of A.
Explanation: We have, $f(x)=x^2-4 x+6$
or $f^{\prime}(x)=2 x-4=2(x-2)$

Therefore, $f^{\prime}(x)=0$ gives $x=2$. Now, the point $x=2$ divides the real line into two disjoint intervals namely, $(-\infty, 2)$ and $(2, \infty)$. In the interval $(-\infty, 2), f ^{\prime}( x )=2 x -4<0$. Therefore, f is strictly decreasing in this interval.
Also, in the interval $(2, \infty), f^{\prime}(x) > 0$ and so the function $f$ is strictly increasing in this interval.
Hence, both the statements are true but Reason is not the correct explanation of Assertion.

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