for the function g(x)=x^5-4x^3 solve the following g(x)>0?

g(x) > 0
x⁵ - 4x³ > 0
x³ (x² - 4) > 0
x³ (x + 2) (x - 2) > 0
x² (x + 2) x (x - 2) > 0

For any real x, x² ≥ 0
Hence, (x + 2) x (x - 2) > 0

Case I : When x < -2
(x + 2) < 0, x < 0 and (x - 2) < 0
Then, (x + 2) x (x - 2) < 0

Case II : When -2 < x < 0
(x + 2) > 0, x < 0 and (x - 2) < 0
Then, (x + 2) x (x - 2) > 0

Case III : When 0 < x < 2
(x + 2) > 0, x > 0 and (x - 2) < 0
Then, (x + 2) x (x - 2) < 0

Case IV : When x > 2
(x + 2) > 0, x > 0 and (x - 2) > 0
Then, (x + 2) x (x - 2) > 0

Cases II and IV fulfill the requirement that (x + 2) x (x - 2) > 0
Hence, range of x : -2 < x < 0 or x > 2

g(x) = x³(x + 2)(x - 2)

Sketch the graph based on this or make a sign chart to find where g is positive.

-Infinity......-2.......0.......2.......Infinity
..............-..........+.......-.......+.........

g(x) > 0 on (-2, 0) U (2, infinity)

x^5 - 4x^3 > 0
Solve x^5-4x^3 = 0
x^3(x^2 -4) = 0
x^2-4 = 0
x= -2, 2
x = 0, 2, -2
t the point on the real line.
--------- --------- ---------- --------
-∞......-2........0......2.......∞

choose one point from each interval and test inequality (1)
(-∞,-2) : choose x=-3
x^5- 4x^3 = -135 < 0 (false)

(-2,0) : choose x= -1
x^5 - 4x^2 = 3 > 0 (true)

(0, 2) : choose x=1 ;
x^5 - 4x^3 = -3 < 0 (false)

(2, ∞) : choose x=5
x^5- 4x^3 = 2625 > 0 (true)

The solution is (-2,0) U (2,∞)
or
0 < x < 2 and x > 2

Begin by finding where g(x) = 0

x^5 - 4x³ = 0

x³(x² - 4) = 0

The roots are x = -2, x = 0, and x = 2.

Check the interval x < -2 by evaluating at x = -3

g(-3) = -135

Check the interval - 2 < x < 0 by evaluating at -1

g(-1) = 3

Check the interval 0 < x < 2 by evaluating at 1:

g(1) = -3

Check the interval x > 2 by evaluating x = 3

g(3) = 135

g(x) > 0 for -2 < x < 0 and x > 2

　
g(x) > 0
x^5 − 4x^3 > 0
x^3 (x^2 − 4) > 0
x^3 (x − 2) (x + 2) > 0

−2 < x < 0 or x > 2

Find the zeroes

x^3 * (x^2 - 4)
x^3 * (x - 2) * (x + 2)

Zeroes at

x = -2 , 0 , 2

So you have 4 intervals to investigate

(-inf , -2) , (-2 , 0) , (0 , 2) , (2 , inf)

Pick values from each interval and see what sign g(x) would have

g(-3) = (-3)^5 - 4 * (-3)^3 = -243 - 4 * (-27) = -243 + 108 = -135
g(-1) = (-1)^5 - 4 * (-1)^3 = -1 + 4 = 3
g(1) = 1^5 - 4 * 1^3 = 1 - 4 = -3
g(3) = 3^5 - 4 * (3)^3 = 243 - 4 * 27 = 243 - 108 = 135