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