Mean Value Theorem and Rolle's Theorem Application

Hi Mustafa, let's solve this calculus problem involving the Mean Value Theorem. We are given a function f that is continuous and differentiable on the interval from 1 to 2, and a helper function g of x.

Let's investigate statement one first. It asks if there exists a point c where the derivative f prime of c equals 2.

Since f is continuous and differentiable, we can apply the Mean Value Theorem. The theorem states there is a c in the interval such that the derivative equals the average rate of change.

Plugging in the given values, we get four minus two over two minus one.

This simplifies to two over one, which is two. So, statement one is definitely true.

Now let's look at statement two. It asks if there is a c such that the derivative of g of x equals zero.

We check the values of g at the endpoints. First, g of one equals f of one divided by one, which is two.

Next, g of two equals f of two divided by two. Substituting the value four for f of two, we get four divided by two, which is also two.

Since g of one equals g of two, and g is continuous and differentiable on the interval, we can apply Rolle's Theorem.

Rolle's Theorem guarantees that there exists at least one c in the open interval such that the derivative of g equals zero. Thus, statement two is true as well.