The curve \(y = \frac{e^x}{\cos x}\), for \(-\frac{1}{2}\pi < x < \frac{1}{2}\pi\), has one stationary point. Find the \(x\)-coordinate of this point.
Solution
To find the stationary point, we need to find the derivative of \(y = \frac{e^x}{\cos x}\) and set it to zero.
Using the quotient rule, the derivative is:
\(\frac{d}{dx} \left( \frac{e^x}{\cos x} \right) = \frac{e^x \cos x + e^x \sin x}{\cos^2 x}\)
Set the derivative to zero:
\(e^x \cos x + e^x \sin x = 0\)
Factor out \(e^x\):
\(e^x (\cos x + \sin x) = 0\)
Since \(e^x \neq 0\), we have:
\(\cos x + \sin x = 0\)
This simplifies to:
\(\tan x = -1\)
The solution for \(x\) in the interval \(-\frac{1}{2}\pi < x < \frac{1}{2}\pi\) is:
\(x = -\frac{1}{4}\pi\)
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