Is #f(x)=e^x+cosx^2# increasing or decreasing at #x=pi/6#?

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Answer 1 · 2015-12-27T00:25:16

Increasing.

Find the sign of #f'(pi/6)#. If it's positive, the function is increasing. If it's negative, the function is decreasing at that point.

Find #f'(x)#.

#f'(x)=d/dx(e^x)+d/dx(cos(x^2))#

#d/dx(e^x)=e^x#

Use the chain rule to find the next derivative.

#d/dx(cos(u))=-u'sin(u)#

So,

#d/dx(cos(x^2))=-2xsin(x^2)#

Thus,

#f'(x)=e^x-2xsin(x^2)#

#f'(pi/6)~~1.405#

Since #f'(pi/6)>0#, the function is increasing when #x=pi/6#.

graph{e^x+cos(x^2) [-10.25, 15.05, -4.4, 8.26]}