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

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

Ex 6.2 Class 12 Maths Question 1.
Show that the function given by f (x) = 3x+17 is strictly increasing on R.
Solution:
f(x) = 3x + 17
∴ f’ (x) = 3>0 ∀ x∈R
⇒ f is strictly increasing on R.

Ex 6.2 Class 12 Maths Question 2.
Show that the function given by f (x) = e2x is strictly increasing on R.
Solution:
We have f (x) = e2x
⇒ f’ (x) = 2e2x
Case I When x > 0, then f’ (x) = 2e2x

Ex 6.2 Class 12 Maths Question 3.
Show that the function given by f (x) = sin x is
(a) strictly increasing in $left( 0,frac { pi }{ 2 } right)$
(b) strictly decreasing in $left( frac { pi }{ 2 } ,pi right)$
(c) neither increasing nor decreasing in (0, π)
Solution:
We have f(x) = sinx
∴ f’ (x) = cosx
(a) f’ (x) = cos x is + ve in the interval $left( 0,frac { pi }{ 2 } right)$
⇒ f(x) is strictly increasing on $left( 0,frac { pi }{ 2 } right)$
(b) f’ (x) = cos x is a -ve in the interval $left( frac { pi }{ 2 } ,pi right)$
⇒ f (x) is strictly decreasing in $left( frac { pi }{ 2 } ,pi right)$
(c) f’ (x) = cos x is +ve in the interval $left( 0,frac { pi }{ 2 } right)$
while f’ (x) is -ve in the interval $left( frac { pi }{ 2 } ,pi right)$
∴ f(x) is neither increasing nor decreasing in (0,π)

Ex 6.2 Class 12 Maths Question 4.
Find the intervals in which the function f given by f(x) = 2x² – 3x is
(a) strictly increasing
(b) strictly decreasing
Solution:
f(x) = 2x² – 3x
⇒ f’ (x) = 4x – 3
⇒ f’ (x) = 0 at x = $frac { 3 }{ 4 }$
The point $x=frac { 3 }{ 4 }$ divides the real

Ex 6.2 Class 12 Maths Question 5.
Find the intervals in which the function f given by f (x) = 2x3 – 3x² – 36x + 7 is
(a) strictly increasing
(b) strictly decreasing
Solution:
f(x) = 2x3 – 3x² – 36x + 7;
f (x) = 6 (x – 3) (x + 2)
⇒ f’ (x) = 0 at x = 3 and x = – 2
The points x = 3, x = – 2, divide the real line into three disjoint intervals viz. (-∞,-2), (-2,3), (3,∞)
Now f’ (x) is +ve in the intervals (-∞, -2) and (3,∞). Since in the interval (-∞, -2) each factor x – 3, x + 2 is -ve.
⇒ f’ (x) = + ve.
(a) f is strictly increasing in (-∞, -2)∪(3,∞)
(b) In the interval (-2,3), x+2 is +ve and x-3 is -ve.
f (x) = 6(x – 3)(x + 2) = + x – = -ve
∴ f is strictly decreasing in the interval (-2,3).

Ex 6.2 Class 12 Maths 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
Solution:
(c) Let f(x) = – 2x3 – 9x2 – 12x + 1
∴ f’ (x) = – 6x2 – 18x – 12
= – 6(x2 + 3x + 2)
f'(x) = – 6(x + 1)(x + 2), f’ (x) = 0 gives x = -1 or x = -2
The points x = – 2 and x = – 1 divide the real line into three disjoint intervals namely ( – ∞, – 2) ( – 2, – 1) and( – 1 ∞).
In the interval (-∞,-2) i.e.,-∞<x<-2 (x+ 1) (x+2) are -ve.
∴f’ (x) = (-) (-) (- ) = – ve.
⇒ f (x) is decreasing in (-∞,-2)
In the interval (-2, -1) i.e., – 2 < x < -1,
(x + 1) is -ve and (x + 2) is + ve.
∴ f'(x) = (-)(-) (+) = + ve.
⇒ f (x) is increasing in (-2, -1)
In the interval (-1,∞) i.e.,-1 <x<∞,(x + 1) and (x + 2) are both positive. f’ (x) = (-) (+) (+) = -ve.
⇒ f (x) is decreasing in (-1, ∞)
Hence, f (x) is increasing for – 2 < x < – 1 and decreasing for x<-2 and x>-1.

Ex 6.2 Class 12 Maths Question 7.
Show that $y=log(1+x)-frac { 2x }{ 2+x } x>-1″>, is an increasing function of x throughout its domain.
Solution:
let

For f (x) to be increasing f’ (x) > 0
$Rightarrow frac { 1 }{ x+1 } >0Rightarrow x>-1″>
Hence,
is an increasing function of x for all values of x > – 1.

Ex 6.2 Class 12 Maths Question 8.
Find the values of x for which y = [x (x – 2)]² is an increasing function.
Solution:
y = x4 – 4x3 + 4x2
$frac { dy }{ dx }$ = 4x3 – 12x2 + 8x
For the function to be increasing $frac { dy }{ dx }$ >0
4x3 – 12x2 + 8x>0
⇒ 4x(x – 1)(x – 2)>0
For 0 < x < 1, $frac { dy }{ dx }$ = (+)(-)(-) = +ve and for x > 2, $frac { dy }{ dx }$ = (+) (+) (+) = +ve
Thus, the function is increasing for 0 < x < 1 and x > 2.

Ex 6.2 Class 12 Maths Question 9.
Prove that $y=frac { 4sintheta }{ (2+costheta ) } -theta$ is an increasing function of θ in $left[ 0,frac { pi }{ 2 } right]$
Solution:
$frac { dy }{ dx } =frac { 8costheta +4 }{ { (2+costheta ) }^{ 2 } } -1=frac { costheta (4-costheta ) }{ { (2+costheta ) }^{ 2 } }$
For the function to be increasing $frac { dy }{ dx }$ > 0
⇒ cosθ(4-cos2θ)>0
⇒ cosθ>0
⇒ θ∈$left[ 0,frac { pi }{ 2 } right]1$

Ex 6.2 Class 12 Maths Question 10.
Prove that the logarithmic function is strictly increasing on (0, ∞).
Solution:
Let f (x) = log x
Now, f’ (x) = $frac { 1 }{ x }$ ; When takes the
values x > 0, $frac { 1 }{ x }$ > 0, when x > 0,
∵ f’ (x) > 0
Hence, f (x) is an increasing function for x > 0 i.e

Ex 6.2 Class 12 Maths Question 11.
Prove that the function f given by f (x) = x² – x + 1 is neither strictly increasing nor strictly decreasing on (-1,1).
Solution:
Given
f (x) = x² – x + 1

∴ f (x) is neither increasing nor decreasing on (-1,1).

Ex 6.2 Class 12 Maths Question 12.
Which of the following functions are strictly decreasing on $left[ 0,frac { pi }{ 2 } right]$
(a) cos x
(b) cos 2x
(c) cos 3x
(d) tan x
Solution:
(a) We have f (x) = cos x
∴ f’ (x) = – sin x < 0 in $left[ 0,frac { pi }{ 2 } right]$
∴ f’ (x) is a decreasing function.

Ex 6.2 Class 12 Maths 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
Solution:
(d) f(x) = x100 + sin x – 1
∴ f’ (x)= 100x99+ cos x
(a) for(-1, 1)i.e.,- 1 <x< 1,-1 <x99< 1
⇒ -100<100x99<100;
Also 0 ⇒ f’ (x) can either be +ve or -ve on(-1, 1)
∴ f (x) is neither increasing nor decreasing on (-1,1).
(b) for (0,1) i.e. 0<x< 1 x99 and cos x are both +ve ∴ f’ (x) > 0
⇒ f (x) is increasing on(0,1)

Ex 6.2 Class 12 Maths 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).
Solution:
We have f (x) = x² + ax + 1
∴ f’ (x) = 2x + a.
Since f (x) is an increasing function on (1,2)
f’ (x) > 0 for all 1 < x < 2 Now, f” (x) = 2 for all x ∈ (1,2) ⇒ f” (x) > 0 for all x ∈ (1,2)
⇒ f’ (x) is an increasing function on (1,2)
⇒ f’ (x) is the least value of f’ (x) on (1,2)
But f’ (x)>0 ∀ x∈ (1,2)
∴ f’ (1)>0 =>2 + a>0
⇒ a > – 2 : Thus, the least value of a is – 2.

Ex 6.2 Class 12 Maths 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.
Solution:
Given
$f(x)=x+frac { 1 }{ x }$

Hence, f’ (x) is strictly increasing on I.

Ex 6.2 Class 12 Maths 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 decreasing on
$left( frac { pi }{ 2 } ,pi right)$
Solution:
f’ (x) = $frac { 1 }{ sinquad x } .cosquad xquad cotquad xquad$
when 0 < x < $frac { pi }{ 2 }$, f’ (x) is +ve; i.e., increasing
When $frac { pi }{ 2 }$ < x < π, f’ (x) is – ve; i.e., decreasing,
∴ f (x) is decreasing. Hence, f is increasing on (0, π/2) and strictly decreasing on (π/2, π).

Ex 6.2 Class 12 Maths 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 $left( frac { pi }{ 2 } ,pi right)$
Solution:
$f(x)=logquad cosx$
f’ (x) = $frac { 1 }{ cosx } (-sinx)=-tanx$
In the interval $left( 0,frac { pi }{ 2 } right)$ ,f’ (x) = -ve
∴ f is strictly decreasing.
In the interval $left( frac { pi }{ 2 } ,pi right)$, f’ (x) is + ve.
∴ f is strictly increasing in the interval.

Ex 6.2 Class 12 Maths Question 18.
Prove that the function given by
f (x) = x3 – 3x2 + 3x -100 is increasing in R.
Solution:
f’ (x) = 3x2 – 6x + 3
= 3 (x2 – 2x + 1)
= 3 (x -1 )2
Now x ∈ R, f'(x) = (x – 1)2≥0
i.e. f'(x)≥0 ∀ x∈R; hence, f(x) is increasing on R.

Ex 6.2 Class 12 Maths Question 19.
The interval in which y = x2 e-x is increasing is
(a) (-∞,∞)
(b) (-2,0)
(c) (2,∞)
(d) (0,2)
Solution:
(d) f’ (x) = 2xe-x + x2( – e-x) = xe-x(2-x) = e-xx(2-x)
Now e-x is positive for all x ∈ R f’ (x) = 0 at x = 0,2
x = 0, x = 2 divide the number line into three disjoint intervals, viz. (-∞, 0), (0,2), (2, ∞)
(a) Interval (-∞,0) x is +ve and (2-x) is +ve
∴ f’ (x) = e-xx (2- x)=(+)(-) (+) = -ve
⇒ f is decreasing in (-∞,0)
(b) Interval (0,2) f’ (x) = e-x x (2 – x)
= (+)(+)(+) = +ve
⇒ f is increasing in (0,2)
(c) Interval (2, ∞) f’ (x) = e-x x (2 – x) = (+) (+) (-)
= – ve
⇒ f is decreasing in the interval (2, ∞)

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