F(x)= X^4 - 8 using analytical methods

Question

To find those values of x which the given function is increasing and those values of x for which it is decreasing 

Answer 1

f(x) = x⁴ - 8

 

f'(x) = 4x³ 

 

f'(x) = 0 when x = 0

 

When x < 0, f'(x)  < 0, so f is decreasing

 

When x > 0, f'(x) > 0, so f is increasing

 

f(x) is decreasing on (-∞, 0) and is increasing on (0,∞).  

Answer 2

You can factor out F(x) using difference of perfect squares.

 

F(x) = (x² - √8)(x² + √8)

 

F(x) = (x - √8)(x + √8)(x² + √8)

 

 

You only have 2 x-intercepts.  Therefore, the 4th degree function acts like a quadratic function.

 

Next, we find the vertex,  Since the formula for the x-coordinate of the vertex is  x=-b/2a,  the vertex is the point (0, -8).

 

Interval of decrease (-∞, 0)

Interval of increase (0, ∞)