Answer
\[\,\,\,\,\,\, - 1,35\]
Work Step by Step
\[\begin{gathered}
f\,\left( x \right) = - {x^4} + 7{x^3} - \frac{{{x^2}}}{2} \hfill \\
Find\,\,the\,\,derivative\,\,of\,\,the\,\,function \hfill \\
{f^,}\,\left( x \right) = \,\,{\left[ { - {x^4} + 7{x^3} - \frac{{{x^2}}}{2}} \right]^,} \hfill \\
Use\,\,the\,\,power\,\,rule \hfill \\
{f^,}\,\left( x \right) = - 4{x^3} + 21{x^2} - x \hfill \\
Find\,\,the\,\,\sec ond\,\,derivative \hfill \\
{f^{\,,,}}\,\left( x \right) = \,\left( { - 4{x^3} + 21{x^2} - x} \right) \hfill \\
{f^{\,,,}}\,\left( x \right) = - 12{x^2} + 42x - 1 \hfill \\
Evaluate\,\,{f^{,,}}\,\left( 0 \right)\,\,and\,{f^{,,}}\,\left( 2 \right)\,\, \hfill \\
{f^{\,,,}}\,\left( 0 \right) = - 12\,{\left( 0 \right)^2} + 42\,\left( 0 \right) - 1 = - 1 \hfill \\
{f^{\,,,}}\,\left( 2 \right) = - 12\,{\left( 2 \right)^2} + 42\,\left( 2 \right) - 1 = 35 \hfill \\
\end{gathered} \]
