How to find points on a graph when given this?

Firstly, the product rule should give the following:

(2x^4 + 1)(1) + (x - 5)(8x^3)

You forgot the exponent on the last part.

You now have to simplify it. This is optional, but makes the rest easier.

(2x^4 + 1) + (8x^4 - 40x^3)

10x^4 - 40x^3 + 1

This is the formula for the slope, and we need to find when it is equal to 1:

10x^4 - 40x^3 + 1 = 1

10x^4 - 40x^3 = 0

x^4 - 4x^3 = 0

x^3 (x - 4) = 0

As we can see, x is equal to either 0 or 4.

We can now plug these into the original function to find the y values:

y = (2[0]^4 + 1) ([0] - 5) = (1) (-5) = -5

y = (2[4]^4 + 1) ([4] - 5) = (513) (-1) = -513

The points where the slope of the tangent line is equal to 0 are (0, -5) and (4, -513)

I hope this helps!