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  2. Let be a polynomial of degree 6 in x, in which the coefficient of x6 is unity and it has extrema at
# Engineering

Let f(x) be a polynomial of degree 6 in x, in which the coefficient of x6 is unity and it has extrema at x=-1\;and\; x=1. If \lim_{x \to0 }\frac{f(x)}{x^{3}}=1\; then 5f(2) is equal to________
Option: 1 12
Option: 2 36
Option: 3 72
Option: 4 144

Answers (1)
S Suraj Bhandari

 

Let f(x)=x^{6}+a x^{5}+b x^{4}+c x^{3}+d x^{2}+e x+f

as \lim _{x \rightarrow 0} \frac{f(x)}{x^{3}}=1  non-zero finite

So, d = e = f = 0

and f(x)=x^{3}\left(x^{3}+a x^{2}+b x+c\right)

Now, as f(x)=x^{6}+a x^{5}+b x^{4}+x^{3}

and \mathrm{f}^{\prime}(\mathrm{x})=0\text{ at }\mathrm{x}=1\text{ and }\mathrm{x}=-1

\begin{aligned} &\text {i.e., } f^{\prime}(x)=6 x^{5}+5 a x^{4}+4 b x^{3}+3 x^{2}\\ &\;\;\;\;\;\;\;\mathrm{f}^{\prime}(1)=0\\ &\Rightarrow 6+5 a+4 b+3=0\\ &\Rightarrow 5 a+4 b=-9 \end{aligned}

\& \\\mathrm{f}^{\prime}(-1)=0 \\ \Rightarrow-6+5 \mathrm{a}-4 \mathrm{~b}+3=0 \\ \Rightarrow 5 \mathrm{a}-4 \mathrm{~b}=3

\begin{aligned} &\text {Solving both we get, }\\ & a=\frac{-6}{10}=\frac{-3}{5} ; \quad b=\frac{-3}{2} \\ & \therefore \quad f(x)=x^{6}-\frac{3}{5} x^{5}-\frac{3}{2} x^{4}+x^{3} \\ & \therefore 5 f(2) =5\left[64-\frac{3}{5} \cdot 32-\frac{3}{2} \cdot 16+8\right] \\ &=320-96-120+40 \\ &=144 \end{aligned}

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