If 8 f(x)+6 f≤ft(1/x)=x+5 and y=x² f(x), then d y/d x at x=-1 equals

Correct Answer is : −114-1/14−141

Correct Answer is : −114-1/14−141

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AP EAPCET 23-Aug-2021 Shift 1 Paper

Section: Mathematics

Question : 66 of 160

Marks: +1, -0

8 f(x)+6 f\left(\frac{1}{x}\right)=x+5

y=x^{2} f(x)

\frac{d y}{d x}

x=-1

equals

Solution:

8 f(x)+6 f\left(\frac{1}{x}\right)=x+5

...(i)

y=x^{2} f(x)

Replace

x

\frac{1}{x}

in Eq. (i), we obtain

8 f\left(\frac{1}{x}\right)+6 f(x)=\frac{1}{x}+5

...(ii)

Solve Eqs. (i) and (ii) for

f(x)

and

f\left(\frac{1}{x}\right)

Multiply by 8 in Eq. (i) and by 6 in Eq. (ii) and subtract, we get

64 f(x)-36 f(x)=8 x-\frac{6}{x}+10

\Rightarrow 28 f(x)=8 x-\frac{6}{x}+10

Now,

28 x^{2} f(x)=8 x^{3}-6 x+10 x^{2}

\Rightarrow 28 y=8 x^{3}-6 x+10 x^{2}

Differentiating it,

28 \frac{d y}{d x}=24 x^{2}-6+20 x

\Rightarrow \frac{d y}{d x}=\frac{24 x^{2}-6+20 x}{28}

\left(\frac{d y}{d x}\right)_{x=-1}

=\frac{24(-1)^{2}-6+20(-1)}{28}

=-\frac{1}{14}

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