Observe the following lists
List - I List - II
A) minimum value of $| x - 5 |$ is 1) -1
B) minimum value of $| x - 3 | + | x - 7 |$ is 2) 0
C) maximum value of $| sin 5 x |$ 3) $\sqrt{2}$
D) maximum value of $| sin + cos x |$ 4) 1
5) 4

A - 2, B - 5, C - 3, D - 2.

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A - 2, B - 5, C - 4, D - 3.

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A - 2, B - 3, C - 1, D - 4.

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A - 2, B - 3, C - 5, D - 1.

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Solution

The correct option is D A - 2, B - 5, C - 4, D - 3.
$A) ∣ x - 5 ∣\geq 0$
So the minimum value of $| x - 5 |$ is $0$
$B) ∣ x - 3 ∣ + ∣ x - 7 ∣$
The minimum value will be at $x = \frac{7 + 3}{2} = 5$
So the minimum value of $∣ x - 3 ∣ + ∣ x - 7 ∣$ is $| 5 - 3 | + | 5 - 7 | = 2 + 2 = 4$
$C) sin 5 x \in [- 1, 1] ⟹ | sin 5 x | \in [0, 1]$
So the maximum value is $1$
$D) sin x + cos x \in [- \sqrt{2}, \sqrt{2}] ⟹ ∣ sin x + cos x ∣\in [0, \sqrt{2}]$
So the maximum value is $\sqrt{2}$

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f (x) = {sin}^{- 1} x + 2 {tan}^{- 1} x + x^{5} - 5 x^{4} + 5 x^{3} + 1