Interval where the function is strictly increasing and decreasing:

To determine the intervals where the function f(x) = 4sin^3(x) - 6sin^2(x) + 12sin(x) + 100 is strictly increasing and decreasing on the interval [0, π], we need to analyze the behavior of its derivative, f'(x).

Finding the derivative:
To find the derivative of f(x), we can differentiate each term separately using the chain rule and sum rule of differentiation.

f'(x) = d/dx(4sin^3(x)) - d/dx(6sin^2(x)) + d/dx(12sin(x)) + d/dx(100)

Taking the derivatives term by term:
f'(x) = 12sin^2(x)cos(x) - 12sin(x)cos(x) + 12cos(x) - 0

Simplifying:
f'(x) = 12sin(x)(sin(x)cos(x) - cos(x) + 1)

Finding the critical points:
To find the critical points, we need to solve the equation f'(x) = 0.

12sin(x)(sin(x)cos(x) - cos(x) + 1) = 0

This equation will be true if either sin(x) = 0 or sin(x)cos(x) - cos(x) + 1 = 0.

Identifying critical points when sin(x) = 0:
When sin(x) = 0, x can take the values 0, π, 2π, and so on. These are the points where the function may change its behavior.

Identifying critical points when sin(x)cos(x) - cos(x) + 1 = 0:
To solve sin(x)cos(x) - cos(x) + 1 = 0, we can factor out cos(x) from the first two terms.

cos(x)(sin(x) - 1) + 1 = 0

Since 1 - sin(x) = cos^2(x), we can rewrite the equation as:

cos(x)cos^2(x) + 1 = 0

cos^3(x) + cos(x) + 1 = 0

Unfortunately, this equation does not have a simple algebraic solution. However, we can use numerical methods or graphs to find the approximate values of x where this equation is satisfied.

Identifying the intervals of increasing and decreasing:
To determine the intervals where the function is strictly increasing or decreasing, we need to analyze the sign of the derivative f'(x) within different intervals.

Interval [0, π/2]:
In this interval, sin(x) and cos(x) are positive, and sin(x)cos(x) - cos(x) + 1 > 0. Therefore, f'(x) > 0.

Interval [π/2, π]:
In this interval, sin(x) is positive, and cos(x) is negative. As sin(x)cos(x) - cos(x) + 1 can be positive or negative depending on the exact values of x, we need to analyze it further.

By plugging in some values, we find that sin(x)cos(x) - cos(x) +