Answer:
[tex]f'(\frac{3\pi}{2})[/tex] is undefined
Step-by-step explanation:
[tex]f(x)=(sinx+1)^x[/tex]
[tex]\frac{d}{dx}(sinx+1)^x[/tex]
[tex]\frac{d}{dx}e^{ln((sinx+1)^x)}[/tex]
[tex]\frac{d}{dx}e^{xln(sinx+1)}[/tex]
[tex](\frac{d}{dx}(x)*ln(sinx+1)+x*\frac{d}{dx}ln(sinx+1))e^{xln(sinx+1)}[/tex]
[tex](ln(sinx+1)+x*cosx(\frac{1}{sinx+1}))(sinx+1)^x[/tex]
[tex][ln(sinx+1)+\frac{xcosx}{sinx+1}](sinx+1)^x[/tex]
[tex]f'(\frac{3\pi}{2})=[ln(sin\frac{3\pi}{2} +1)+\frac{\frac{3\pi}{2} cos\frac{3\pi}{2} }{sin\frac{3\pi}{2} +1}](sin\frac{3\pi}{2} +1)^{\frac{3\pi}{2}}[/tex]
[tex]f'(\frac{3\pi}{2})=[ln(-1+1)+\frac{\frac{3\pi}{2} (0) }{-1+1}](-1+1)^{-1}[/tex]
[tex]f'(\frac{3\pi}{2})=[ln(0)+\frac{0}{0}](0)^{-1}[/tex]
Because the derivative is undefined, then the function isn't differentiable at the point [tex](\frac{3\pi}{2},0)[/tex], making it a critical point since the slope of the function is 0.
