Is f(x)=cotx-e^xtanxf(x)=cotxextanx increasing or decreasing at x=pi/6x=π6?

1 Answer
Feb 7, 2016

The function is decreasing.

Explanation:

To figure the increasing or decreasing nature of a function we can look at its derivative. If f'(x) is positive at the specified x value then the function is increasing otherwise if f'(x) is negative at the specified value then the function is decreasing.

If the f'(x)=0 then we have a stationary point.

So, to differentiate the function:

f'(x) = -csc^2(x) - e^xtanx-e^xsec^2(x)

The product rule was used in the differentiation.

Now putting our value of x into our function:

f'(pi/6)= -csc^2(pi/6)-e^(pi/6)tan(pi/6)-e^(pi/6)sec^2(pi/6)

=-4-e^(pi/6)(1/3+4/3)=-4-5/4e^(pi/6)<0

Therefore as f'(pi/6) is negative our function must be decreasing as we can see from the graph of f(x) below.

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