NCERT Solutions for Class 12 Maths Chapter 6 Application of Derivatives — Exercise 6.2

Question 1:Show that the function given by f (x) = 3x + 17 is increasing on R.

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Question 2:Show that the function given by f (x) = e2x is increasing on R.

Question 3:Show that the function given by f (x) = sin x is (a) increasing in 0, 2 π   (b) decreasing in , 2 π  p (c) neither increasing nor decreasing in (0, p)

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Question 4:Find the intervals in which the function f given by f (x) = 2x2 – 3x is (a) increasing (b) decreasing

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Question 5:Find the intervals in which the function f given by f (x) = 2x3 – 3x2 – 36x + 7 is (a) increasing (b) decreasing

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Question 6:Find the intervals in which the following functions are strictly increasing or decreasing: (a) x2 + 2x – 5 (b) 10 – 6x – 2x2 (c) –2x3 – 9x2 – 12x + 1 (d) 6 – 9x – x2 (e) (x + 1)3 (x – 3)3

Question 7:Show that 2 log(1 ) 2 x y x x = + − + , x > – 1, is an increasing function of x throughout its domain.

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Question 8:Find the values of x for which y = [x(x – 2)]2 is an increasing function.

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Question 9:Prove that 4sin (2 cos ) y q = − q + q is an increasing function of q in 0 2 , é p ë ê ù û ú .

Question 10:Prove that the logarithmic function is increasing on (0, ∞).

Question 11:Prove that the function f given by f (x) = x2 – x + 1 is neither strictly increasing nor decreasing on (– 1, 1).

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Question 12:Which of the following functions are decreasing on 0, 2 p ? (A) cos x (B) cos 2x (C) cos 3x (D) tan x

1. Show that the function given by f (x) = 3x + 17

Question 13:On which of the following intervals is the function f given by f (x) = x100 + sin x –1 decreasing ? (A) (0,1) (B) , 2 p π (C) 0, 2 p (D) None of these

Question 14:For what values of a the function f given by f (x) = x2 + ax + 1 is increasing on [1, 2]?

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Question 15:Let I be any interval disjoint from [–1, 1]. Prove that the function f given by 1 f (x) x x = + is increasing on I.

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Question 16:Prove that the function f given by f (x) = log sin x is increasing on 0 2 , π æè ç öø ÷ and decreasing on p p 2 , æè ç öø ÷ .

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Question 17:Prove that the function f given by f (x) = log |cos x| is decreasing on 0, 2 π   and increasing on 3 , 2 2 π  p .

Question 18:Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.

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Question 19:The interval in which y = x2 e–x is increasing is (A) (– ∞, ∞) (B) (– 2, 0) (C) (2, ∞) (D) (0, 2)