Determine the intervals on which each function f is increasing or decreasing?

Question

Use a sign chart to determine the intervals on which each function f is increasing or decreasing. Then verify your algebraic answers with graphs

 

1. F(x) = x^3 + 4x^2 + 4x -5

 

2. F(x) = 3x + 1/ x^2 - 1

Answer 1

(1).

The function F(x) = x³ + 4x² + 4x - 5.

Differentiate with respect to x.

F'(x) = 3x² + 8x + 4

To find the critical numbers of F(x), set F'(x) = 0 or F'(x) does not exist.

3x² + 8x + 4 = 0

3x² + 6x + 2x + 4 = 0

3x(x + 2) + 2(x + 2) = 0

(3x + 2)(x + 2) = 0

3x + 2 = 0 and x + 2 = 0

x = - 2/3 = 0 and x = -2.

The critical numbers are x = - 2/3 = 0 and x = -2 and the test intervals are (-∞, -2), (-2, -2/3) and (-2/3, ∞).

Interval     Test Value                     Sign of F'(x)                     Conclusion

(-∞, -2)        x = -3       F'(-3) = 3(-3)² + 8(-3) + 4 = 7 > 0     Increasing

(-2, -2/3)     x = -1       F'(-1) = 3(-1)² + 8(-1) + 4 = -1 < 0    Decreasing

(-2/3, ∞)     x = 0         F'(0) = 3(0)² + 8(0) + 4 = 4 > 0         Increasing

Graph :

Answer 2

(2).

The function F(x) = 3x + 1/x² - 1.

Differentiate with respect to x.

F'(x) = 3 - 2/x³

F'(x) = (3x³ - 2)/x³

To find the critical numbers of F(x), set F'(x) = 0 or F'(x) does not exist.

F'(x) = 0 ⇒ 3x³ - 2 = 0 ⇒ x = (2/3)^1/3 = 0.874.

The function F'(x) does not exist when x = 0.

 

The imaginary numbers are negligible, The critical numbers are x = 0 and x = 0.874 and the test intervals are (-∞, 0), (0, 0.874) and (0.874, ∞).

Interval	Test Value	Sign of F'(x)	Conclusion
(-∞, 0)	x = -1	F'(-1) = 3 - 2/(-1)³ = 5 > 0	Increasing
(0, 0.874)	x = 0.1	F'(0.1) = 3 - 2/(0.1)³ = - 1997 < 0	Decreasing
(0.874, ∞)	x = 1	F'(1) = 3 - 2/(1)³ = 1 > 0	Increasing

Graph :