Question 5
Let the f : R → R be defined by f (x) = 2x + cos x, then f :
(A) has a minimum at x = 𝜋 (B) has a maximum, at x = 0
(C) is a decreasing function (D) is an increasing function
Given
f (𝒙) = 2𝒙 + cos 𝒙
Since we need to check
If it is increasing or decreasing
Or, at what value of 𝑥 the function has maximum or minimum value
Thus, we check sign of 𝒇′(𝒙)
∴ Differentiating 𝑓(𝑥) w.r.t. 𝑥
f (𝑥) = 2𝑥 + cos 𝑥
f’ (𝒙) = 2 − sin 𝒙
Checking sign of 𝒇′(𝒙)
We know that
−1 ≤ sin 𝒙 ≤ 1
Multiplying by −1
−1 × (−1) ≥ −sin 𝑥 ≥ 1 × (−1)
1 ≥ − sin 𝑥 ≥ −1
−1 ≤ − sin 𝒙 ≤ 1
Adding 2 both sides
2 − 1 ≤ 2 − sin 𝑥 ≤ 2 + 1
1 ≤ 2 − sin 𝒙 ≤ 3
1 ≤ 𝒇′(𝒙)≤ 3
∴ 𝒇′(𝒙) > 0
Thus, f (𝑥) is increasing function
So, the correct answer is (D)

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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