The minimum value of the slope of the curve y=x^5+x^3−2x is: a. 0
b. 2
c. 6
d. -2
e. none of these.

The given curve is --- y = x^5 + x^3 - 2x 

First derivative to this curve is y' = 5 x^4 + 3 x^2 - 2 

=> Slope (m) = 5 x^4 + 3 x^2 - 2 

For the minimum value of m we calculate dm/dx and put it = 0 

=> d( 5 x^4 + 3 x^2 - 2 ) / dx = 0 

=> 20 x^2 + 6 x = 0 

=> x ( 20 x + 6 ) = 0 

Turning values of x are 0 and - 6 / 20 = ( - 3 / 10 ) 

At x = 0 , m = - 2 

and at x = - 3/10 

m = 5 x^4 + 3 x^2 - 2 

=> m = 5 ( - 3 / 10 )^4 + 3 ( - 3 / 10 )^2 - 2 = - 1.68 

Hence at turning points, the slope is minimum at x = 0 and is equal to = - 2 

MINIMUM VALUE OF THE SLOPE = - 2

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