Question

Let f(x) be a continuously differentiable function on the interval (0, ∞) such that f(1) = 2 and \(\lim\limits_{t→x}\frac{t^{10}f(x)-x^{10}f(t)}{t^9-x^9}=1\)
for each x > 0. Then, for all x > 0, f(x) is equal to

• A. \(\frac{31}{11x}-\frac{9}{11}x^{10}\)
• B. \(\frac{9}{11x}+\frac{13}{11}x^{10}\)
• C. \(\frac{-9}{11x}+\frac{31}{11}x^{10}\)
• D. \(\frac{13}{11x}+\frac{9}{11}x^{10}\)

Answer 1

The Correct Option is B

To solve the problem, we need to analyze the given limit condition and find the function \( f(x) \) satisfying it.

1. Understanding the limit expression:
Given:
\[ \lim_{t \to x} \frac{t^{10} f(x) - x^{10} f(t)}{t^9 - x^9} = 1 \] for each \( x > 0 \).

2. Rewrite the limit to use the derivative concept:
Notice the numerator and denominator both tend to zero as \( t \to x \). This suggests using the definition of derivative or applying algebraic manipulation.

3. Rewrite numerator and denominator:
\[ \frac{t^{10} f(x) - x^{10} f(t)}{t^9 - x^9} = \frac{t^{10} f(x) - x^{10} f(x) + x^{10} f(x) - x^{10} f(t)}{t^9 - x^9} = \frac{f(x)(t^{10} - x^{10}) - x^{10} (f(t) - f(x))}{t^9 - x^9} \]

4. Separate the fraction:
\[ = f(x) \frac{t^{10} - x^{10}}{t^9 - x^9} - x^{10} \frac{f(t) - f(x)}{t^9 - x^9} \]

5. Analyze the first fraction as \( t \to x \):
Since \( t \to x \), using derivative formulas:
\[ \lim_{t \to x} \frac{t^{10} - x^{10}}{t^9 - x^9} \] Both numerator and denominator tend to zero, apply L'Hôpital's Rule:
Derivative of numerator w.r.t. \( t \): \( 10 t^9 \)
Derivative of denominator w.r.t. \( t \): \( 9 t^8 \)
Thus, the limit is:
\[ \frac{10 x^9}{9 x^8} = \frac{10}{9} x \]

6. Analyze the second fraction as \( t \to x \):
\[ \lim_{t \to x} \frac{f(t) - f(x)}{t^9 - x^9} = \lim_{t \to x} \frac{f(t) - f(x)}{t - x} \cdot \frac{t - x}{t^9 - x^9} = f'(x) \cdot \lim_{t \to x} \frac{1}{9 t^8} = \frac{f'(x)}{9 x^8} \]

7. Substitute limits back:
\[ 1 = f(x) \times \frac{10}{9} x - x^{10} \times \frac{f'(x)}{9 x^8} = \frac{10}{9} x f(x) - \frac{x^2}{9} f'(x) \]

8. Multiply both sides by 9:
\[ 9 = 10 x f(x) - x^{2} f'(x) \]

9. Rearranged differential equation:
\[ x^{2} f'(x) + 9 = 10 x f(x) \] or \[ x^{2} f'(x) - 10 x f(x) = -9 \]

10. Divide both sides by \( x^2 \):
\[ f'(x) - \frac{10}{x} f(x) = -\frac{9}{x^2} \] This is a linear differential equation of the form:
\[ f'(x) + P(x) f(x) = Q(x) \] with \( P(x) = -\frac{10}{x} \) and \( Q(x) = -\frac{9}{x^2} \).

11. Find the integrating factor (IF):
\[ IF = e^{\int P(x) dx} = e^{\int -\frac{10}{x} dx} = e^{-10 \ln x} = x^{-10} \]

12. Multiply entire equation by integrating factor:
\[ x^{-10} f'(x) - \frac{10}{x} x^{-10} f(x) = -\frac{9}{x^2} x^{-10} \] which simplifies to \[ \frac{d}{dx} \left( x^{-10} f(x) \right) = -9 x^{-12} \]

13. Integrate both sides:
\[ \int \frac{d}{dx} \left( x^{-10} f(x) \right) dx = \int -9 x^{-12} dx \] \[ x^{-10} f(x) = -9 \int x^{-12} dx + C \] \[ = -9 \times \left( \frac{x^{-11}}{-11} \right) + C = \frac{9}{11} x^{-11} + C \]

14. Solve for \( f(x) \):
\[ f(x) = x^{10} \left( \frac{9}{11} x^{-11} + C \right) = \frac{9}{11} x^{-1} + C x^{10} \]

15. Use the initial condition \( f(1) = 2 \):
\[ f(1) = \frac{9}{11} (1)^{-1} + C (1)^{10} = \frac{9}{11} + C = 2 \] \[ C = 2 - \frac{9}{11} = \frac{22}{11} - \frac{9}{11} = \frac{13}{11} \]

Final form of \( f(x) \):
\[ \boxed{ f(x) = \frac{9}{11x}  + \frac{13}{11} x^{10} } \]

Answer 2

To solve this problem, let's analyze the given equation: \(\lim\limits_{t→x}\frac{t^{10}f(x)-x^{10}f(t)}{t^9-x^9}=1\).

We understand that this is a form of derivative. Rewrite the expression in a more recognizable derivative form:

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

Notice that:

\(\frac{t-x}{t^9-x^9}\) = \(\frac{1}{(t^{9}+t^{8}x+t^{7}x^{2}+\ldots+x^{9})}\)

So, the limit becomes:

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

Set the equation equal to 1 and solve:

\(9x^{9}f(x) - x^{10}f'(x) = 1\).

From here, rearrange the equation to find f(x):

\(x^{10}f'(x)/f(x) = 9x^{9} - 1\).

Integrating both sides:

\(\ln |f(x)| = \int(9x^{-1} - x^{-10})dx\).

Evaluating the integral:

\(\ln |f(x)| = 9\ln(x) - \frac{x^{-9}}{-9} + C\).

Thus, the general solution is:

\(|f(x)| = e^{9\ln(x) + x^{-9}/9 + C}\).

Apply given \(f(1) = 2\) into the equation:

\(f(1) = e^{9\ln(1) + 1^{-9}/9 + C} = 2\).

Simplify the equation:

\(e^{C + \frac{1}{9}} = 2\).

Solve for C:

\(C = \ln(2) - \frac{1}{9}.\)

Substitute back into the equation f(x) and get:\(\frac{9}{11x}+\frac{13}{11}x^{10}\).

This matches the correct answer provided.