#### Solved Examples and Worksheet for Finding Intervals for Increasing and Decreasing Functions

Q1For all x (0, ∞), f(x) = e7x-e- 7x2 is
A. not continuous
B. decreasing
C. increasing
D. stationary

Step: 1
Here f(x) = e7x-e- 7x2 is continuous in (0, ∞)
[Write the function.]
Step: 2
f′(x) = ddx (e7x-e-7x2)
[Differentiate.]
Step: 3
= 7e7x+7e- 7x2
Step: 4
For all x (0, ∞), 72(e7x+e- 7x) > 0
[Solve the inequality.]
Step: 5
So, f ′ (x) > 0
Step: 6
So, f(x) is increasing in (0, ∞).
Q2Find the interval in which y = 4x3 - 42x2 + 144x + 27 decreases.
A. (- ∞, ∞)
B. (3, 4)
C. (- ∞, 3)
D. (4, ∞)

Step: 1
Here y = 4x3 - 42x2 + 144x + 27
[Given function.]
Step: 2
dydx = ddx (4x3 - 42x2 + 144x + 27)
[Differentiate.]
Step: 3
= 12x2 - 84x + 144
Step: 4
= 12(x2 - 7x + 12)
Step: 5
For x2 - 7x + 12 < 0, dydx < 0
Step: 6
For (x - 3)( x - 4) < 0, dydx < 0
[Factorize.]
Step: 7
For x (3, 4), dydx < 0
[Solve the inequality.]
Step: 8
So, y decreases for all x (3, 4)
Correct Answer is :   (3, 4)
Q3Which of the following is the largest interval in which y = 9x2 + ln |x| decreases?

A. (9, ∞)
B. (- ∞, ∞)
C. (- ∞, 0)
D. (18, ∞)

Step: 1
y = 9x2 + ln|x|
[Given function.]
Step: 2
dydx = ddx (9x2 + ln|x|)
[Differentiate.]
Step: 3
= 18x + 1x
Step: 4
= 18x2 + 1x
Step: 5
For 18x2 + 1x < 0, dydx < 0
Step: 6
For 1x < 0, dydx < 0
[18x2 + 1 > 0 for all x ≠ 0.]
Step: 7
For x < 0, dydx < 0
[Solve the inequality.]
Step: 8
So y decreases for all x (- ∞, 0)
Correct Answer is :   (- ∞, 0)
Q4Which of the following is the largest interval in which y = - tan-13x decreases ?
A. (- ∞,∞)
B. (- ∞, 0)
C. (3 , ∞)
D. (0, ∞)

Step: 1
y = - tan-13x
[Write the function.]
Step: 2
dydx = - ddx(tan-13x)
[Differentiate.]
Step: 3
= - 31+9x2
Step: 4
For all real values of x, - 31 + 9x2 < 0
[For a decreasing function.]
Step: 5
So, dydx < 0
Step: 6
So, y = - tan-13x decreases in (- ∞,∞).
Correct Answer is :   (- ∞,∞)
Q5Find the interval in which f(x) = (x - 6)2 + (x - 10)2 decreases.

A. (-∞, ∞)
B. (8, ∞)
C. (- ∞, 8)
D. (- 8, 8)

Step: 1
Here f(x) = (x - 6)2 + (x - 10)2
[Given function.]
Step: 2
f ′ (x) = 2(x - 6) + 2(x - 10)
[Differentiate.]
Step: 3
= 2(2x - 16)
Step: 4
= 4(x - 8)
Step: 5
For x - 8 < 0, f ′ (x) < 0
[Simplify.]
Step: 6
For x < 8, f ′ (x) < 0
Step: 7
So f(x) decreases in (- ∞, 8)
Correct Answer is :   (- ∞, 8)
Q6In the interval (0, π2), f(x) = cos (sin x) is

A. decreasing
B. stationary
C. increasing
D. not differentiable

Step: 1
Here f(x) = cos (sin x) is a differential function in (0, π2)
[Given function.]
Step: 2
f ′ (x) = ddxcos (sin x)
[Differentiate.]
Step: 3
= - sin (sin x) ddx (sin x)

Step: 4
= - sin (sin x)cos x

Step: 5
For x (0, π2), sinx (0, 1) and cos x (0, 1)
Step: 6
=> sin (sin x) > 0 and cos x > 0
Step: 7
sin (sin x) cos x > 0
Step: 8
- sin (sin x) cos x < 0
Step: 9
So f ′ (x) < 0 for x (0, π2)
Step: 10
So in (0, π2), f(x) is decreasing.
Q7In the interval (-3π4, -π4), f(x) = sinx - cos x is

A. stationary
B. not differentiable
C. decreasing
D. increasing

Step: 1
Here f(x) = sin x - cos x is differentiable in(-3π4, -π4)
Step: 2
f ′ (x) = ddx (sin x - cos x)
[Differentiate.]
Step: 3
= cos x + sinx

Step: 4
= 2(12cos x +12 sinx)
[Multiply and divide by 2.]
Step: 5
= 2 (sinπ4cos x + cosπ4sinx)
Step: 6
= (2)sin(π4 + x)
Step: 7
For x (-3π4, -π4),π4 + x (-π2, 0)
Step: 8
sin (π4 + x) < 0
Step: 9
2sin(π4 + x) < 0
Step: 10
f ′ (x) < 0
Step: 11
So in (-3π4, -π4), f(x) decreases.
Q8In (e, ∞), the function y = x3x is

A. decreasing
B. not differentiable
C. increasing
D. stationary

Step: 1
y = x3x is differentiable in (e, ∞)
[Write the function.]
Step: 2
ln y = 3x ln x
[Take ln on both sides.]
Step: 3
ddx (lny) = ddx (3x ln x)
Step: 4
1ydydx = 3x2 -3ln xx2
[Use product rule.]
Step: 5
= 3-3ln xx2
Step: 6
dydx = y(3-3ln xx2)
Step: 7
= x3x(3-3ln xx2)
Step: 8
In (e, ∞), 3 - 3ln x < 0, so dydx < 0
Step: 9
So, in (e, ∞) y is decreasing.
Q9For all real values of x, f(x) = - sinh (sinhx) is

A. decreasing
B. increasing
C. not differentiable
D. stationary

Step: 1
Here f(x) = - sinh (sinh x) is differentiable for all real x.
Step: 2
f ′ (x)= -ddx(- sinh (sinh x))
[Differentiate.]
Step: 3
= - cosh (sinh x) ddx (sinh x)

Step: 4
= - cosh (sinh x) cosh x

Step: 5
For all real values of x, cosh (sinhx) coshx > 0
[cosh x >0 for all real x.]
Step: 6
So (- cosh (sinh x) cosh x) < 0 for all x
Step: 7
f′ (x) < 0 for all real values of x
Step: 8
So for all real values of x, f(x) is decreasing.
Q10Find the interval in which y = x2 + 11x + 20 increases.

A. (- 112, 0) (0, ∞)
B. (- 2, 0) (0, ∞)
C. (112, ∞)
D. (- 2, ∞)
E. (- 112, ∞)

Step: 1
y = x2 + 11x + 20
[Write the function.]
Step: 2
dydx = ddx(x2 + 11x + 20)
[Find dydx.]
Step: 3
= 2x + 11
[Use Sum Rule.]
Step: 4
y increases if dydx > 0
Step: 5
dydx > 0 if 2x + 11 > 0
Step: 6
x > - 112
[Solve the inequality.]
Step: 7
The interval in which y increases is (- 112, ∞).
Correct Answer is :   (- 112, ∞)
Q11Find the interval in which f(x) = ex(3 + x) increases.
A. (- 3, 0) (0, ∞)
B. (- 3, ∞)
C. (0, ∞)
D. (- 4, ∞)
E. For all real values of x

Step: 1
f(x) = ex(3 + x)
[Write the function.]
Step: 2
f ′(x) = ddx [ex(3 + x)]
[Find f ′(x).]
Step: 3
f ′ (x) = ex(1) + (3 + x) ex
[Use Product Rule.]
Step: 4
= ex(4 + x)
[Factor.]
Step: 5
f(x) increases if f ′ (x) > 0
Step: 6
f ′ (x) > 0 if ex(4 + x) > 0
Step: 7
x > - 4
[ex is positive for all x.]
Step: 8
The interval in which f(x) = ex(3 + x) increases is (- 4, ∞).
Correct Answer is :   (- 4, ∞)
Q12In which interval does the function f(x) = x2-3x-2x+3 increase?
A. (73, ∞)
B. (- ∞, - 7) (1, ∞)
C. (- 7, 1)
D. (- 3, ∞)
E. (- ∞, - 7] [1, ∞)

Step: 1
f(x) = x2-3x-2x+3
[Write the function.]
Step: 2
f ′(x) = ddx (x2-3x-2x+3)
[Find f ′(x).]
Step: 3
f ′ (x) = (x+3)(2x-3)-(x2-3x-2)(1)(x+3)2
[Use Quotient Rule.]
Step: 4
= 2x2-3x+6x-9-x2+3x+2(x+3)2
Step: 5
= x2+6x-7(x+3)2 = (x-1)(x+7)(x+3)2
[Simplify.]
Step: 6
f(x) increases if f ′ (x) > 0
Step: 7
f ′ (x) > 0 if (x-1)(x+7)(x+3)2 > 0
Step: 8
x < - 7 or x > 1
Step: 9
The interval in which f(x) increases is (- ∞, - 7) (1, ∞).
Correct Answer is :   (- ∞, - 7) (1, ∞)
Q13Find the interval in which f(x) = (x2+49)12 increases.

A. (0, ∞)
B. (- ∞, 0)
C. (7, ∞)
D. (- 12, ∞)
E. (- 7, ∞)

Step: 1
f(x) = (x2+49)12
[Write the function.]
Step: 2
f ′(x) = ddx [(x2+49)12]
[Find f ′(x).]
Step: 3
= 12 (x2+49)12-1(2x)
[Use Power Rule.]
Step: 4
= x(x2+49)12
Step: 5
f(x) increases if f ′ (x) > 0
Step: 6
f ′ (x) > 0 if x(x2+49)12 > 0
Step: 7
x > 0
[(x2+49)12 > 0 for all x.]
Step: 8
The interval in which f(x) = (x2+49)12 increases is (0, ∞).
Correct Answer is :   (0, ∞)
Q14Which of the following is true for f(x) = 5x + e7x?
A. f(x) decreases for all x R
B. f(x) increases for all x (0, ∞)
C. f(x) increases for all x (- ∞, 0)
D. f(x) increases for all x R
E. f ′ (x) decreases for all x R

Step: 1
f(x) = 5x + e7x
[Write the function.]
Step: 2
f ′(x) = ddx (5x + e7x)
[Find f ′(x).]
Step: 3
f ′ (x) = 5 + 7e7x
[Use Sum Rule.]
Step: 4
f(x) increases if f ′ (x) > 0
Step: 5
f ′ (x) > 0 if 5 + 7e7x > 0
Step: 6
So, f(x) increases for all x R.
[e7x > 0 for all x.]
Correct Answer is :   f(x) increases for all x R
Q15Find the interval in which y = x4 - 9x3 decreases.

A. (- ∞, 0) (0, 427)
B. (0, 274)
C. (- ∞, 0) (0, 9)
D. (- ∞, 0) (0, 274)
E. (- ∞, 0) (0, 9)

Step: 1
y = x4 - 9x3
[Write the function.]
Step: 2
dydx = ddx(x4 - 2x3)
[Find dydx.]
Step: 3
= 4x3 - 27x2
[Use Difference Rule.]
Step: 4
= x2[4x - 27]
[Factor.]
Step: 5
y decreases if dydx < 0
Step: 6
dydx < 0 if x2(4x - 27) < 0
Step: 7
x (- ∞, 0) (0, 274)
[Solve the inequality.]
Step: 8
So, the function y = x4 - 9x3 decreases for all x (- ∞, 0) (0, 274).
Correct Answer is :   (- ∞, 0) (0, 274)