Question

How to find the values of X

Answer 1

The cubic equation y=x^3-3x^2+1 can be differentiated to obtain a formula for the slope of any tangent line to the curve. 

Thid derivative is dy/dx = 3x^2-6x.

Since we're interested in finding x values at which the tangent line is horiz., we set this deriv. = to 0.  Factoring, 3x(x-2)=0.  Thus, x=0 and x=2.

You could verify this by graphing the original function on a graphing calculator.  Determine from your graph the x-values at which the tangent lines to the graph appear to be horiz.