Critical point of exponential functions and trig functions,
Let's see some examples that don't just involve powers of x.
Example: find out all the critical points for the function.
y = 6x - 4 cos (3x )
Solution : Firstly get the derivative and don't forget to utilize the chain rule on the second term.
y′ = 6 + 12 sin (3x )
Now, it will exist everywhere and therefore there won't be any critical points for which the derivative doesn't present. The only critical points will come from points which make the derivative zero. We will have to solve,
6 + 12 sin (3x ) = 0
sin (3x ) = - 1/2
Solving out this equation gives the following.
3x = 3.6652 + 2 ∏ n, n = 0, ±1, ±2,......
3x = 5.7596 +2 ∏ n, n = 0, ±1, ±2,......
Now divide by 3 to obtain all the critical points for this function.
x = 1.2217 + (2 ∏ n /3), n = 0, ±1, ±2,......
x = 1.9199 + (2 ∏ n/3 ), n = 0, ±1, ±2,......