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

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

f(x) = x4 - 8
 
f'(x) = 4x
 
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,∞).  
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, ∞)

Comments

 
(x2 - √8) = (x - √√8)(x + √√8)