## To solve the equation 2sin^2x = sqrt(x), we need to find the exact solutions on the interval [0, pi].

Let's break down the problem step by step:

Step 1: Rewrite the equation using trigonometric identities.

We know that sin^2x = (1/2)(1-cos2x) and sqrt(x) = x^(1/2). So, we can rewrite the given equation as:

2(1/2)(1-cos2x) = x^(1/2)

Simplifying further:

1 - cos2x = x^(1/2)

Step 2: Solve for cos2x.

Rearranging the equation:

1 - x^(1/2) = cos2x

Using the identity cos(2Î¸) = 1 - 2sin^2(Î¸), we can substitute 2x in place of Î¸:

cos2x = 1 - 2sin^2(x)

Step 3: Substitute cos2x in the equation.

1 - 2sin^2(x) = 1 - x^(1/2)

Rearranging further:

2sin^2(x) = x^(1/2)

Step 4: Set up the equation for solving.

Now, we have:

2sin^2(x) = x^(1/2)

Step 5: Solve the equation using calculus.

To find the exact solutions on the interval [0, pi], we can use calculus techniques:

Make the observation that both sides are equal to zero at x = 0.

Differentiate both sides of the equation:

d/dx (2sin^2(x)) = d/dx (x^(1/2))

4sin(x)cos(x) = (1/2)x^(-1/2)

Solve for x using the derivative:

4sin(x)cos(x) = (1/2)x^(-1/2)

This equation does not have an analytical solution, so numerical methods or technology (like a graphing calculator or software) may be used to approximate the solutions.

In summary, the equation 2sin^2x = sqrt(x) can be solved by rewriting it, and using calculus techniques to find the exact solutions on the interval [0, pi].