Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a thrice differentiable odd function satisfying $f^{\prime}(x) \geq 0, f^{\prime\prime}(x)=f(x), f(0)=0, f^{\prime}(0)=3$. Then $9 f\left(\log _e 3\right)$ is equal to __________ .
Answer (integer)
36
Solution
<p>$$\begin{aligned}
&f^{\prime}(x) \geq 0, f^{\prime \prime}(x)=f(x)\\
&\text { Second order differential equation }
\end{aligned}$$</p>
<p>$$\begin{aligned}
& f(x)=A e^x+B e^{-x} \\
& f(0)=0 \Rightarrow A=-B \\
& \Rightarrow f(x)=A\left(e^x-e^{-x}\right) \\
& f^{\prime}(x)=A e^x+A e^{-x}=A\left(e^x+e^{-x}\right) \\
& f^{\prime}(0)=3=A\left(e^0+e^{-0}\right)=2 A \Rightarrow A=\frac{3}{2} \\
& f(x)=\frac{3}{2}\left(e^x-e^{-x}\right) \\
& \text { If }(\ln 3)=\frac{27}{2}\left(e^{\ln 3}-e^{-\ln 3}\right)=\frac{27}{2}\left(3-\frac{1}{3}\right)=\frac{27}{2} \cdot \frac{8}{3} \\
& =36
\end{aligned}$$</p>
About this question
Subject: Mathematics · Chapter: Differentiation · Topic: Derivatives of Standard Functions
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