Final answer:
The critical numbers of the function h(x) = sin²(x) * cos(x) are found by setting its derivative equal to zero. After using the product and chain rules for differentiation, and solving the trigonometric equation, the critical numbers obtained are π/4, 3π/4, 5π/4, and 7π/4.
Step-by-step explanation:
To find the critical numbers of the function h(x) = sin²(x) * cos(x) for 0 < x < 2π, you need to find the values of x where the derivative of h(x) is zero or undefined.
First, take the derivative of h(x) using the product rule and the chain rule:
h'(x) = d/dx [sin²(x) * cos(x)]
h'(x) = 2sin(x)cos(x) * cos(x) - sin²(x) * sin(x)
h'(x) = 2sin(x)cos²(x) - sin³(x)
Setting h'(x) to zero gives us:
2sin(x)cos²(x) - sin³(x) = 0
sin(x)(2cos²(x) - sin²(x)) = 0
There are two scenarios for when this equation equals zero:
- sin(x) = 0, which has no solutions in the interval (0, 2π)
- 2cos²(x) - sin²(x) = 0, which simplifies to cos²(x) = sin²(x)/2 or cos(x) = ±sin(x)/√2
For cos(x) = sin(x)/√2, we get x = π/4 and 5π/4. For cos(x) = -sin(x)/√2, we get x = 3π/4 and 7π/4. All of these are within the specified interval.
Thus, the critical numbers of the function h(x) are π/4, 3π/4, 5π/4, and 7π/4.