Mathematics - Application of Derivative Question with Solution | TestHub

$\begin{array}{rl}& \underset{x\to 0}{\lim }\frac{f(x)}{x^2}=5\\ & \underset{x\to 0}{\lim }\frac{a{x}^4+b{x}^3+c{x}^2+dx+e)}{x^2}=5\\ & c=5\text{ and }d=e=0\\ & f(x)=a{x}^4+b{x}^3+5x^2\\ & f^{\prime }(x)=4a{x}^3+3b{x}^2+10x\\ & =x\left(4a{x}^2+3bx+10\right)\end{array}$
has extremes at 4 and so $f^{\prime }(4)=0\&f^{\prime }(5)=0$
so $\mathrm{a}=\frac{1}{8}\&\mathrm{b}=\frac{-3}{2}$
$\text{ so }f(2)=\frac{1}{8}\times 2^4-\frac{3}{2}\times 2^3+5\times 2^2$
$=2-12+20=10$