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  • Let f(x) be a polynomial of degree 5 such that  are its critical points. If <img alt="\lim_{x\rightarrow 0}\left ( 2+\frac{f(x)}{x^{3}} \right )=4," src="

Let f(x) be a polynomial of degree 5 such that x=\pm 1 are its critical points. If \lim_{x\rightarrow 0}\left ( 2+\frac{f(x)}{x^{3}} \right )=4, then which one of the following is not true ?
Option: 1 f(1)-4(-1)=4.  
Option: 2 x=1 is a point of maxima and x=-1 is a point of maxima and x=-1  
Option: 3 f is an odd function.
Option: 4  x=1 is a point of minima and x=-1 is a point of maxima of f.
 

Answers (1)

best_answer

 

 

Algebraic Function of type ‘infinity (∞) -

Algebraic Function of type ‘infinity (∞)’

To find the limit of the type  x ? ∞, write the given expression in the form of  \frac{N}{D}(D ≠ 0), (N is numerator, D is denominator). Then divide both N and D by highest power of x occurring in both N and D to get a meaningful form.

An important result:

\\\text { If } m, n \text { are positive integers and } a_{0}, b_{0} \neq 0 \text { are non-zero real numbers, then }\\\\\lim _{x \rightarrow \infty} \frac{a_{0} x^{m}+a_{1} x^{m-1}+\ldots+a_{m-1} x+a_{m}}{b_{0} x^{n}+b_{1} x^{n-1}+\ldots+b_{n-1} x+b_{n}}=\begin{cases} 0& \text{ if } \;\;\;m<n \\ \frac{a_0}{b_0}& \text{ if } \;\;m=n \\ \infty & \text{ if } \;\;m>n \;\;\;when\;\;a_0b_0>0\\ -\infty & \text{ if } \;\;m>n\;\;\;when\;\;a_0b_0<0 \end{cases}

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{f(x)=a x^{5}+b x^{4}+c x^{3}} \\ {\lim _{x \rightarrow 0}\left(2+\frac{a x^{5}+b x^{4}+c x^{3}}{x^{3}}\right)=4}\\ {\Rightarrow 2+c=4 \Rightarrow c=2} \\ {f^{\prime}(x)=5 a x^{4}+4 b x^{3}+6 x^{2}} \\ {=x^{2}\left(5 a x^{2}+4 b x+6\right)}

\begin{array}{l}{f^{\prime}(1)=0 \quad \Rightarrow \quad 5 a+4 b+6=0} \\ {f^{\prime}(-1)=0 \quad \Rightarrow \quad 5 a-4 b+6=0}\end{array}

On solving b=0 \text{ and } a=-\frac{6}{5}

\begin{array}{l}{f(x)=\frac{-6}{5} x^{5}+2 x^{3}} \\ {f^{\prime}(x)=-6 x^{4}+6 x^{2}} \\ {=6 x^{2}\left(-x^{2}+1\right)} \\ {=-6 x^{2}(x+1)(x-1)}\end{array}

f(x) is maxima at x = 1 and minima at x = –1

Correct Option (4)

Posted by

Ritika Jonwal

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