Question

Does Sin(ln(x))/sqrt(e^x+5) have a absolute min on 1 12 years ago

Answer 1

Start off by looking for the function's critical points. When is its derivative zero or undefined?
\[f(x)=\frac{\sin\ln x}{\sqrt{e^x+5}}~~\Rightarrow~~f'(x)=\frac{\dfrac{\sqrt{e^x+5}}{x}\cos\ln x-\dfrac{e^x\sin\ln x}{2\sqrt{e^x+5}}}{e^x+5}\]
\[\frac{\dfrac{\sqrt{e^x+5}}{x}\cos\ln x-\dfrac{e^x\sin\ln x}{2\sqrt{e^x+5}}}{e^x+5}=\frac{\dfrac{1}{x}\cos\ln x-\dfrac{e^x\sin\ln x}{2(e^x+5)}}{\sqrt{e^x+5}}\]
Clearly, \(f'\) is undefined for \(x\le0\).

Now, given such an intricate function, it's not inconceivable to think you can use a graphing utility to find the critical points.