Let f(x) be a continuously differentiable function on the interval (0, ∞) such that f(1) = 2 and

Question

Let \(f(x)\) be a continuously differentiable function on the interval \((0, \infty)\) such that \(f(1)=2\) and \(\lim _{t \rightarrow x} \frac{t^{10} f(x)-x^{10} f(t)}{t^{9}-x^{9}}=1\)

for each \(x>0\). Then, for all \(x>0, f(x)\) is equal to

(A) \(\frac{31}{11 x}-\frac{9}{11} x^{10}\)

(B) \(\frac{9}{11 x}+\frac{13}{11} x^{10}\)

(C) \(\frac{-9}{11 x}+\frac{31}{11} x^{10}\)

(D) \(\frac{13}{11 x}+\frac{9}{11} x^{10}\)

Answer 1

Correct option is (B) \(\frac{9}{11 x}+\frac{13}{11} x^{10}\)

\(\lim\limits_{t \rightarrow x} \frac{t^{10} f(x)-x^{10} f(t)}{t^{9}-x^{9}}=1\)

\(\lim\limits_{t \rightarrow x} \frac{10t^9f(x) - f'(t)x^{10}}{9t^8}=1\)

\( \Rightarrow 10 x^{9} f(x)-f(x) x^{10}=9 x^{8} \)

\(\Rightarrow f^{\prime}(x)-\frac{10}{x} f(x)=-\frac{9}{x^{2}}\)

\( { IF }=e^{-\int \frac{10}{x} d x}=\frac{1}{x^{10}}\)

\(\therefore \text {Sol} ^n\)

\(\frac{y}{x^{10}}=\int-\frac{9}{x^{10}} \times \frac{1}{x^{2}} d x \)

\( =-9 \int x^{-12} d x\)

\(\frac{y}{x^{10}}=\frac{9}{11} x^{-11}+C\)

\(\because y(1)=2 \Rightarrow C=\frac{13}{11}\)

\(\Rightarrow y=\frac{9}{11 x}+\frac{13}{11} x^{10}\)