# 2nd PUC Maths Question Bank Chapter 6 Application of Derivatives Ex 6.2

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## Karnataka 2nd PUC Maths Question Bank Chapter 6 Application of Derivatives Ex 6.2

### 2nd PUC Maths Application of Derivatives NCERT Text Book Questions and Answers Ex 6.2

Question 1.
Find the intervals in which the function f given by f (x) = 3x + 17 is strictly increasing in R
f (x) = 3x + 17
f’ (x) = 3 > 0 , ∀ x ∈ R
hence f(x) is strictly increasing on R.

Question 2.
Show that the function given by f (x) = e2x is strightly increasing on R
f’ (x) = 2 e2x > 0 ∀ x ∈ R
hence f (x) is strictly increasing on R. Question 3.
Show that the function given by f(x) = sin x is
(a) strictly increasing in $$(0, \pi / 2)$$
(b) strictly decreasing in $$(\pi / 2, \pi)$$ Question 4.
Find the intervals in which the function f given by f (x) = 2x2 – 3x is
(a) strictly increasing
(b) strictly decreasing Question 5.
Find the intervals in which the function f given by f (x) = 2x2 – 3x2 – 36x +7 is
(a) strictly increasing
(b) strictly 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
f (x) = (x + 1)3 (x – 3)3
f’ (x) = (x + 1)3 x 3(x – 3)2 + (x – 3)3 x 3 (x + 1)2
= (x + 1)2 (x – 3)2 [3x + 3 + 3x – 9]  Question 7.
Show that $$y=\log (1+x)-\frac{2 x}{2+x}, x>-1$$ 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 $$y=\frac{4 \sin \theta}{(2+\cos \theta)}-\theta$$ is an a increasing function of θ in $$\left[0, \frac{\pi}{2}\right]$$ Question 10.
Prove that the logarithmic function is strictly increasing on (0, ∞). Question 11.
Prove that the function f given by f (x) = x2 – x + 1 is neither strictly increasing nor strictly decreasing on (- 1, 1).  Question 12.
Which of the following function are strictly decreases on $$(0, \pi / 2)$$
(a) cos x (b) f(x) = 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 strictly decreasing ?
(A) (0,1)
(B) $$\left(\frac{\pi}{2}, \pi\right)$$
(C) $$\left(0, \frac{\pi}{2}\right)$$
(D) None of these Question 14.
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on (1, 2).
f (x) = x2 + ax + 1
f’ (x) = 2x + a
f (x) is increasing if f’ (x) > 0
2x + a > 0 is x > – a/2
2x > -a – a < 2x ⇒ a > – 2x
since x ∈ (1,2) a > -2
The least value is -2. Question 15.
Let I be any interval disjoint from (-1, 1). Prove that the function f given by $$f(x)=x+\frac{1}{x}$$ is strictly increasing on I. Question 16.
Prove that the function f given by f (x) = log sin x is strictly increasing on
$$\left(0, \frac{\pi}{2}\right)$$ and strictly increasing on $$\left(0, \frac{\pi}{2}\right)$$and strictly decreasing on $$\left(\frac{\pi}{2}, \pi\right)$$ Question 17.
Prove that the function f given by f (x) = log cos x is strictly decreasing on
$$\left(0, \frac{\pi}{2}\right)$$and strictly increasing on $$\left(\frac{\pi}{2}, \pi\right)$$ Question 18.
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
f (x) = x3 – 3x2 + 3x
f'(x) = 3x2 – 6x + 3
= 3 (x – 2)2
f'(x)>0 ∀ x ∈ R
∴ function is continuous on R Question 19.
The interval in which y = x2 e-x is increasing is
(A) (- ∞ , ∞)
(B) ( – 2, 0)
(C) (2, ∞)
(D) (0,2). 