Find max and min values of y=sinx sin2x

y = sinx sin2x

y' = cosx sin2x + 2sinx cos2x
= 2sinx cos^2x + 2sinx - 4sin^3x
= 2sinx - 2sin^3x + 2sinx - 4sin^3x
= 4sinx - 6sin^3x
= 2sinx (2 - 3sin^2x)

so, max/min occurs when sinx = √(2/3)
When six = √(2/3), cosx = √(1/3)
so, sin2x = √(8/9)

max/min of y is thus ±√(16/27) = 4√3/9 = .7698

To find the maximum and minimum values of the function y = sin(x) * sin(2x), we can follow these steps:

Step 1: Find the critical points of the function by determining where the derivative of y equals zero or is undefined.

Let's differentiate y with respect to x:

dy/dx = (cos(x) * sin(2x)) + (2sin(x) * cos(2x))

To find the critical points, we need to solve the equation dy/dx = 0:

(cos(x) * sin(2x)) + (2sin(x) * cos(2x)) = 0

Step 2: Solve the equation for x to find the critical points.

There isn't a simple algebraic solution for this equation, so we can use numerical methods or graphing software to estimate the values of x that satisfy the equation.

Step 3: Evaluate the function at the critical points and the endpoints of the interval to find the maximum and minimum values of y.

Once we have the critical points, we can calculate the corresponding values of y by substituting those values into the original function y = sin(x) * sin(2x).

Additionally, we need to evaluate the function at the endpoints of the interval to consider the possibility of the maximum or minimum values occurring at those points.

Step 4: Compare the values of y to identify the maximum and minimum values.

After gathering the values of y at the critical points and endpoints, we can compare them to determine the maximum and minimum values of the function.

Note: Since the given function is periodic (with a period of 2π), we only need to consider the interval from 0 to 2π.