1. Show that the function given by f (x) = 3x + 17 is increasing on R. 2. Show that the function given by f (x) = e2x is increasing on R. 3. Show that the function given by f (x) = sin x is (a) increasing in 0, 2 p (b) decreasing in , 2 p p (c) neither increasing nor decreasing in (0, p) 4. Find the intervals in which the function f given by f (x) = 2x2 3x is (a) increasing (b) decreasing 5. Find the intervals in which the function f given by f (x) = 2x3 3x2 36x + 7 is (a) increasing (b) decreasing 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 7. Show that 2 log(1 ) 2 x y x x = + + , x >
Question 1:Show that the function given by f (x) = 3x + 17 is increasing on R.
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)
Question 4:Find the intervals in which the function f given by f (x) = 2x2 – 3x is (a) increasing (b) decreasing
Question 5:Find the intervals in which the function f given by f (x) = 2x3 – 3x2 – 36x + 7 is (a) increasing (b) decreasing
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.
Question 8:Find the values of x for which y = [x(x – 2)]2 is an increasing function.
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).
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
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]?
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.
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 ÷ .
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.
Question 19:The interval in which y = x2 e–x is increasing is (A) (– ∞, ∞) (B) (– 2, 0) (C) (2, ∞) (D) (0, 2)
