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Discuss the curve $$y=x^{4}-4 x^{3}$$ with respect to concavity, points of inflection
and local maximum and minimum.Use this information to sketch the curve

$$f(x)=x^{4}-4 x^{3}$$
$$f^{\prime}(x)=4 x^{3}-12 x^{2}$$
$$f^{\prime \prime}(x)=12 x^{2}-24 x$$

(1) To find Critical numbarse $$\rightarrow f^{\prime}(x)=0$$

$$4 x^{3}-12 x^{2}=0 \rightarrow 4 x^{2}(x-3)=0$$

$$x=0, x=3$$

$$f(0)=0^{4}-4(0)^{3}=0$$ local min
$$f(3)=(3)^{4}-4(3)^{3}=-27$$

$$f^{\prime \prime}(0)=0$$
$$f^{\prime \prime}(3)=36>0$$
$$f^{\prime}(3)=0$$

\ $$f(3)=-27$$ is alocal min

$$f^{\prime \prime}(x)=12 x^{2}-24 x$$

$$12 x^{2}-24 x=0 \longrightarrow 12 x(x-2)=0$$

$$x=0 \quad,\quad x=2$$

$$(0,0)$$ Inflection point
$$(2,-16)$$ Inflection point

$$(3,-27)$$
$$(0,0)$$
$$(2,-16)$$

 

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