Two statements are given, one labelled Assertion (A) and the other labelled Reason (R).
Let $f (x) = x^4 - 2x^2 + 5$ be defined on [−2, 2].
Assertion (A): The range of $f(x)$ is [2, 13].
Reason (R): The greatest value of $f$ is attained at $x = 2$.
Select the correct answer from the options given below.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → Assertion (A) is false, but Reason (R) is true.
$f(x) = x^2 - 2x^2+5⇒ f'(x) = 4x^3 - 4x$
$⇒f'(x) = 4x (x-1) (x + 1)$
$⇒f'(x) = 4x (x^2 - 1) ⇒ f'(x) = 4x (x − 1) (x + 1)$.
For critical points, $f'(x) = 0⇒ x = 0, −1, 1$.
Now, $f(-2)=(-2)^4-2(-2)^2+5=16-8+5=13$
$f(2) = 2^4-2(2)^2+5=16-8+5=13$
$f(-1)=(-1)^4-2(-1)^2+5=1-2+5=4$
$f(0) = 0-2 × 0+5=5$
$f(1) = 1^4 - 2(1)^2 + 5 = 4$.
So, the range of $f$ is [4, 13]
∴ Assertion is false.
Also, $f$ attains it maximum value at $x = -2$ and $x = 2$
∴ Reason is true.