Describe very specifically the physical appearance of the line tangent to y=5x^6+6x^5-7 at x=-1, and explain why you are describing it in this way

Question

Calculus 

Answer 1

To get the slope of the line first take the derivative of y with respect to x and factor

 

y'=30x^5+30x^4=30x^4(x+1)

 

This derivative equals zero at

 

        x =0 and x =-1

 

So x=-1 is an extrema of y. This extrema is either a minimum or a maximum. The second derivative tells us which. 

 

y''=150x^4+120x^3

 

y''(-1)=30>0

 

Suppose f(x) is a function of x that is twice differentiable at a stationary point x_0.

1. If f^('')(x_0)>0, then f has a local minimum at x_0.

2. If f^('')(x_0)<0, then f has a local maximum at x_0.

 

So x=-1 is a local minimum

 

The line slope, m=y'(-1)=0

 

evaluating the function y at the point of interest

 

y(-1)=5-6-7=-8

 

so the line tangent to the function at the local minimum -1 is the horizontal line y=-8