Question

# Determine all significant features (approximately if necessary) and sketch a graph. $f(x)=2 x^{4}-11 x^{3}+17 x^{2}$

Solution

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Step 1
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$f(x)=2x^{4}-11x^{3}+17x^{2}=x^{2}(2x^{2}-11x+17)$

defined for all reals.$f(0)=0$

$f^{\prime}(x)=8x^{3}-33x^{2}+34x$

$=x(8x^{2}-33x+34)$

Using a calculator to find the zeros,

The critical numbers are $x=0, x=2$ and $x=2.125$. Testing signs,

$\left[\begin{array}{llllllllll} & -\infty & & 0 & & 2 & & 2.125 & & \infty\\ test.p. & & -1 & & 1 & & 2.01 & & 3 & \\ f^{\prime}(x) & & - & 0 & + & 0 & - & 0 & + & \\ f(x) & & \searrow & min & \nearrow & MAX & \searrow & min & \nearrow & \end{array}\right]$

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