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**1. Homework Statement**

Determine how many real numbers satisfy the equation

**x**. Give reasons for your answer naming any theorems you use.

^{3}-6x^{2}+1=0**2. The attempt at a solution**

let:f(x)=x

^{3}-6x

^{2}+1

→f'(x)=3x

^{2}-12x

for stationary points:f'(x)=0

→3x(x-4)=0

∴x=0 or x=4

to determine if minimum or maximum points:

f''(x)=6x-12

→f''(0)=-12<0

→f''(4)=12>0

∴(0,1) is a maximum point.

∴(4,-31) is a minimum point.

Not sure how to finish. I know it has 3 solutions, but I'm not sure as how to express my answer in the correct format. Help?

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