What are the critical points of #f (x) = e^x + ln(6x^2+x)#?
An approximate answer is
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
#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.