What are the critical points of #f (x) = e^x + ln(6x^2+x)#?
1 Answer
An approximate answer is
Explanation:
To find the critical points, we need to compute the first derivative. Since the derivative of a sum is the sum of the derivatives, we can split the problem in two subproblems:

The derivative of
#e^x# is simply#e^x# itself, so the first term is easy to solve 
As for
#ln(6x^2+6x)# , we need to use the chain rule: we have to differentiate the outer function, and then multiply for the derivative of the inner function. The outer function is a logarithm, and so its derivative is the inverse of the argument, which is#1/(6x^2+6x)# . This must be multiplied by the derivative of#6x^2+6x# , which is#12x+6#
Now we have to sum the two terms to obtain the derivative:
The critical points are the zeroes of the derivative, so we should solve
but this is a trascendental equation, so the best you can do is asking a calculator for an approximate value of the solution, as for example here.