Questions: Find the number c that satisfies the conclusion of the Mean Value Theorem f(x)=sqrt(x), [0,4] c=

Solution Steps

To find the number $c$ that satisfies the conclusion of the Mean Value Theorem for the function $f(x) = \sqrt{x}$ on the interval $[0, 4]$, we need to follow these steps:

1. Verify that the function $f(x) = \sqrt{x}$ is continuous on the closed interval $[0, 4]$ and differentiable on the open interval $(0, 4)$.
2. Calculate the average rate of change of the function over the interval $[0, 4]$, which is $\frac{f(4) - f(0)}{4 - 0}$.
3. Find the derivative of the function, $f'(x)$.
4. Set the derivative equal to the average rate of change and solve for $c$.

The function $f(x) = \sqrt{x}$ is continuous on the closed interval $[0, 4]$ and differentiable on the open interval $(0, 4)$. Therefore, the conditions of the Mean Value Theorem are satisfied.

The average rate of change of $f$ over the interval $[0, 4]$ is given by: $$\text{Average Rate of Change} = \frac{f(4) - f(0)}{4 - 0} = \frac{\sqrt{4} - \sqrt{0}}{4 - 0} = \frac{2 - 0}{4} = \frac{1}{2}$$

The derivative of the function $f(x) = \sqrt{x}$ is: $$f'(x) = \frac{1}{2\sqrt{x}}$$

We set the derivative equal to the average rate of change: $$\frac{1}{2\sqrt{c}} = \frac{1}{2}$$ Solving for $c$: $$\sqrt{c} = 1 \implies c = 1$$

Final Answer