Finding Intervals for Increasing and Decreasing Functions-Algebra1-Solved Examples

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

Solutions

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, ∞).

Correct Answer is : increasing

Q2Find the interval in which y = 4x³ - 42x² + 144x + 27 decreases.
A. (- ∞, ∞)
B. (3, 4)
C. (- ∞, 3)
D. (4, ∞)

Solutions

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, ∞)

Solutions

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, ∞)

Solutions

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)

Solutions

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

Solutions

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.

Correct Answer is : decreasing

Q7In the interval (-3π4, -π4), f(x) = sinx - cos x is

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

Solutions

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.

Correct Answer is : decreasing

Q8In (e, ∞), the function y = x3x is

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

Solutions

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.

Correct Answer is : decreasing

Q9For all real values of x, f(x) = - sinh (sinhx) is

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

Solutions

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.

Correct Answer 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, ∞)

Solutions

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) = e^x(3 + x) increases.
A. (- 3, 0) ∪ (0, ∞)
B. (- 3, ∞)
C. (0, ∞)
D. (- 4, ∞)
E. For all real values of x

Solutions

Step: 1

f(x) = e^x(3 + x)

[Write the function.]

Step: 2

f ′(x) = ddx [e^x(3 + x)]

[Find f ′(x).]

Step: 3

f ′ (x) = e^x(1) + (3 + x) e^x

[Use Product Rule.]

Step: 4

= e^x(4 + x)

[Factor.]

Step: 5

f(x) increases if f ′ (x) > 0

Step: 6

f ′ (x) > 0 if e^x(4 + x) > 0

Step: 7

x > - 4

[e^x is positive for all x.]

Step: 8

The interval in which f(x) = e^x(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, ∞)

Solutions

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

[Solve the quadratic inequality.]

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, ∞)

Solutions

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 + e^7x?
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

Solutions

Step: 1

f(x) = 5x + e^7x

[Write the function.]

Step: 2

f ′(x) = ddx (5x + e^7x)

[Find f ′(x).]

Step: 3

f ′ (x) = 5 + 7e^7x

[Use Sum Rule.]

Step: 4

f(x) increases if f ′ (x) > 0

Step: 5

f ′ (x) > 0 if 5 + 7e^7x > 0

Step: 6

So, f(x) increases for all x ∈ R.

[e^7x > 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)

Solutions

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)

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

Related Worksheet

Related Topic

HighSchool Math