To find the critical numbers of a function, the first step is to take its derivative. The derivative of a function represents the slope at any given point, and the critical numbers occur where the slope is zero or undefined. These points often represent local maximum or minimum points, or points where the function changes direction.
So, let's start by calculating the first derivative of h(x) = cos(2x) + 2sin(x).
Using the chain rule (which states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function) and the derivatives of the sine and cosine functions, we have:
h'(x) = -2sin(2x) + 2cos(x)
At the critical points, this derivative is either zero or undefined.
Now, we're going to solve the equation h'(x) = 0 for x:
-2sin(2x) + 2cos(x) = 0
From the solution, we get three potential critical points: π/6, π/2, 5π/6.
Note: These values are obtained from solving the above equation, which involves techniques like translating trigonometric functions to complex exponentials, algebraic factoring, and complex exponentials back to trigonometric functions.
Yet, we only want the critical points within the interval [0, π]. So let's filter out points not on this interval.
On checking, all three solutions π/6, π/2, 5π/6 fall into the interval [0, π].
Hence, the critical numbers of the function h(x) = cos(2x) + 2sin(x) in the interval [0, π] are π/6, π/2, 5π/6.