Question

Determine all significant features (approximately if necessary) and sketch a graph. f(x)=2x411x3+17x2f(x)=2 x^{4}-11 x^{3}+17 x^{2}

Solution

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f(x)=2x411x3+17x2=x2(2x211x+17)f(x)=2x^{4}-11x^{3}+17x^{2}=x^{2}(2x^{2}-11x+17)

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

f(x)=8x333x2+34xf^{\prime}(x)=8x^{3}-33x^{2}+34x

=x(8x233x+34)=x(8x^{2}-33x+34)

Using a calculator to find the zeros,

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

[022.125test.p.112.013f(x)0+00+f(x)minMAXmin]\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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