Question:medium
If $f(x)=x^2+g^{\prime}(1) x+g^{\prime \prime}(2)$ and $g(x)=f(1) x^2+x f^{\prime}(x)+f^{\prime \prime}(x)$, then the value of $f(4)-g(4)$ is equal to ______
If $f(x)=x^2+g^{\prime}(1) x+g^{\prime \prime}(2)$ and $g(x)=f(1) x^2+x f^{\prime}(x)+f^{\prime \prime}(x)$, then the value of $f(4)-g(4)$ is equal to ______
Updated On: Mar 31, 2026
Show Solution
Correct Answer: 14
Solution and Explanation
Computing derivatives: \[ f'(x) = 2x + g'(1), \quad f''(x) = 2 \] Solving for \( g(x) \), \[ g(x) = f(1)x^2 + x[2x + g'(1)] + 2 \] \[ g'(x) = 2f(1)x + 4x + g'(1) \] \[ g''(x) = 2f(1) + 4 \] Setting boundary conditions: \[ f(1) = -2, \quad g'(1) = -3 \] \[ f(x) = x^2 - 3x \] \[ g(x) = -3x + 2 \] \[ f(4) - g(4) = 14 \]
Was this answer helpful?
0
Top Questions on Logarithmic Differentiation
- If $ y = \ln(x^2 + 1) $, then find $ \frac{dy{dx} $ at $ x = 1 $.
- BITSAT - 2025
- Mathematics
- Logarithmic Differentiation
- If $ f(x) = e^{2x} \sin x $, find $ f'(x) $.
- BITSAT - 2025
- Mathematics
- Logarithmic Differentiation
- If \( x^y = e^{x-y} \), prove that \(\frac{dy}{dx} = \frac{\log x}{(1 + \log x)^{2}}\).
- CBSE Class XII - 2024
- Mathematics
- Logarithmic Differentiation
- For what value of \( k \), the function given below is continuous at \( x = 0 \)?
\[ f(x) = \begin{cases} \frac{\sqrt{4} + x - 2x}{x}, & x \neq 0 \\ k, & x = 0 \end{cases} \]- CBSE Class XII - 2024
- Mathematics
- Logarithmic Differentiation
- Want to practice more? Try solving extra ecology questions todayView All Questions
