Find the minimum value?

Question

Find the minimum value of the slope of the curve y=x^5+x^3-2x.

 

Answer: -2

 

Please show all your work. 

Answer 1

See answer provided by Richard P, as modified with my comment ... yes, the answer (minimum slope) is -2.

Answer 2

The slope is the derivative of the function.

 

f'  = 5 x⁴ + 3 x²  -2

 

This factors as ( 5 x²   -2 )  ( x² +1)

 

The minimum is obtained by setting  the first factor to zero.  So x² = 2/5   and

 

x =  ± sqrt(2/5)