Question

Consider the function h(x)= 3x/π +cos x. Since h(x) is continuous on [0, π /2 ] and differentiable on (0, π /2 ), the Mean Value Theorem applies. Find x=c, 0 , that satisfies the conditions of the Mean Value Theorem. Show the work that leads to your answer.

Answer 1

The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists a value c in (a, b) such that the derivative of the function at c is equal to the average rate of change of the function over the interval [a, b]. To find the value of x=c, we need to find the derivative of the function h(x) and solve for x=c.

Step-by-step explanation:

The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists a value c in (a, b) such that the derivative of the function at c is equal to the average rate of change of the function over the interval [a, b].

In this case, the function h(x) = (3x/π) + cos(x) is continuous on [0, π/2] and differentiable on (0, π/2). Therefore, we can apply the Mean Value Theorem to find a value c in (0, π/2) where the derivative of h(x) is equal to the average rate of change of h(x) over [0, π/2].

To find such a value c, we need to find the derivative of h(x) and then solve for x=c.