Help needed with Calculus

Question

Determine if the Mean Value Theorem for Integrals applies to the function f(x) = 2 - x² on the interval  . If so, find the x-coordinates of the point(s) guaranteed by the theorem.

Options:

1)	No, the Mean Value Theorem for Integrals does not apply
2)	Yes,
3)	Yes,
4)	Yes,

Answer 1

The Mean Value Theorem for Integrals states that if f(x) is a continuous function on the closed interval [a, b], then there exists a number c in that interval such that:

∫ f(x)dx = f(c)*(b-a) (The integral is a definite integral from a lower value a to an upper value b)

Since f(x) = 2 - x² is continuous on the interval [0, √2] we can apply the Mean Value Theorem for Integrals.

F(x) = ∫ f(x)dx = 2x - x³/3

Evaluate the integral at x=√2 and then at x=0, then take the difference

F(√2) = 2√2 - 2√2/3 = 4√2/3

F(0) = 0

Therefore the integral is equal to F(√2) - F(0) = 4√2/3

We plug this into the MVT for integrals and then solve for c

4√2/3 = (√2 - 0)*f(c)

4√2/3 = √2*(2-c²)

4/3 = 2-c²

c² = 2/3

c = √(2/3)