What are the values and types of the critical points, if any, of $f(x)=x^4 - 8x^3 + 18x^2 - 27$ ?

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Answer 1 · 2018-07-17T15:24:29

$x=3,f(3)=0$ is an inflection point. and for $(0;-27)$ a minimum point.

Your $f(x)$ is equal to

$f(x)=(x+1)(x-3)^3$

so

$f'(x)=x(4x^2-24x+36)$
so

$x=0$ or $x=3$

$f''(x)=12x^2-48x+36$

$f''(0)=36>0$ we get a minimum

$f''(3)=0$

$f'''(x)=24x-48$

$f'''(3)=48ne 0$

since the derivative has an odd order we have an inflection point.

We have

$(0,-27)$ a minimum

and for

$(3;0)$ an inflection point.

What are the values and types of the critical points, if any, of f(x)=x^4 - 8x^3 + 18x^2 - 27f(x)=x^4 - 8x^3 + 18x^2 - 27?