Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a polynomial function of degree four having extreme values at $x=4$ and $x=5$.
If $\lim _{x \rightarrow 0} \frac{f(x)}{x^2}=5$, then $f(2)$ is equal to :

Question

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a polynomial function of degree four having extreme values at $x=4$ and $x=5$. If $\lim _{x \rightarrow 0} \frac{f(x)}{x^2}=5$, then $f(2)$ is equal to :

Aicharya · Accepted Answer

$\begin{aligned} & \lim _{x ightarrow 0} \frac{f(x)}{x^2}=5 \ & \lim _{x ightarrow 0} \frac{\left.a x^4+b x^3+c x^2+d x+eight)}{x^2}=5 \ & c=5 	ext { and } d=e=0 \ & f(x)=a x^4+b x^3+5 x^2 \ & f^{\prime}(x)=4 a x^3+3 b x^2+10 x \ & =x\left(4 a x^2+3 b x+10ight) \end{aligned}$ has extremes at 4 and so $f^{\prime}(4)=0 \& f^{\prime}(5)=0$ so $\mathrm{a}=\frac{1}{8} \& \mathrm{~b}=\frac{-3}{2}$ $	ext { so } f(2)=\frac{1}{8} 	imes 2^4-\frac{3}{2} 	imes 2^3+5 	imes 2^2$ $=2-12+20=10 $

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a polynomial function of degree four having extreme values at…

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