Find the absolute max of f(x)=5x-e^(5x)
to calculate the max of any function (absolute or relative) u must derivate it and find the value of x that cancels that derivative. for your case, here is the solution :
*** Solution ***
let's derive f(x):
f'(x) = 5 - 5*e^(5x)
condition that we impose : f'(x) = 0
thus : 5 - 5*e^(5x) = 0
5 = 5*e^(5x) ==> e^(5x) = 1
in order to extract the value of x, we must apply a neperian logarithm to both sides, therefore :
ln (e^(5x)) = ln 1 ==> 5x = 0 ==> x = 0
the absolute max of f(x) occurs for x = 0 and leads to f(0) = -1 (when u replace x = 0 in f(x))