The Maximized Slope

Calculus Level 5

Let function \(f\) be defined as \(f(x)=(\cos(a))^x+(\sin(a))^x\) where \(a\) is a parameter that is constantly changing within the interval \(0<a<\frac{\pi}{2}\).

The \(y\)-intercept's tangent line is maximized when the value of \(a\) reaches a certain value.

If the value of the maximized slope is expressed as \(\ln(\)\(\frac{m}{n}\)\()\) where \(m\) and \(n\) are coprime integers, find the value of \(m+n\).

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