Q: Find the minimum value of $\displaystyle 5 \mathrm{cosh} x + 3 \mathrm{sinh} x$.

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So I know that to find the minimum value, I have to differentiate the equation. Then make it equal to zero and find the value of $\displaystyle x$ by rearranging the equation to make $\displaystyle x$ the subject so I did that ... but got the wrong answer.

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Method:

Let $\displaystyle y = 5 \mathrm{cosh} x + 3 \mathrm{sinh} x$

$\displaystyle \therefore \frac{\mathrm{d}y}{\mathrm{d}x} = 5 \mathrm{sinh} x + 3 \mathrm{cosh} x$

$\displaystyle 5 \mathrm{sinh} x + 3 \mathrm{cosh} x = 0$

$\displaystyle 5(e^x - e^{-x})\frac{1}{2} = -3(e^x + e^{-x})\frac{1}{2}$

$\displaystyle 5e^x - 5e^{-x} = -3e^x - 3e^{-x}$

$\displaystyle 8e^x - 2e^x = 0$

$\displaystyle 8e^{2x} - 2 = 0$

$\displaystyle 8e^{2x} = 2$

$\displaystyle e^{2x} = \frac{1}{4}$

$\displaystyle 2x = \ln \left( \frac{1}{4} \right)$

$\displaystyle x = \frac{1}{2} \ln \left( \frac{1}{4} \right)$.

...That is not the correct answer. The correct answer is $\displaystyle 4$ so where have I gone wrong? Thanks in advance.