Apply the Mean Value Theorem and find a value of c in the interval [4,9] that satisfies the equation f(x)=x−−√ . (1 point)
To apply the Mean Value Theorem for this function, we need to:
1. Verify that the function is continuous on the interval [4, 9].
2. Verify that the function is differentiable on the interval (4, 9).
1. The function f(x) = √x is continuous on the interval [4, 9] since the function √x is continuous for all non-negative values of x.
2. To check if the function is differentiable on the interval (4, 9), we need to calculate the derivative of f(x). The derivative of f(x) = √x is given by:
f'(x) = 1/(2√x)
The function 1/(2√x) is continuous and defined for all values of x in the interval (4, 9), except x = 0. Since 0 is not included in the interval (4, 9), the function is differentiable on this interval.
So, the conditions for applying the Mean Value Theorem are satisfied.
According to the Mean Value Theorem, there exists a value c in the interval (4, 9) such that:
f'(c) = (f(9) - f(4))/(9 - 4)
Let's calculate:
f(9) = √9 = 3
f(4) = √4 = 2
So, the equation becomes:
f'(c) = (3 - 2)/(9 - 4)
= 1/5
Now, let's find c:
f'(c) = 1/5
1/(2√c) = 1/5
Cross-multiplying:
2√c = 5
√c = 5/2
Squaring both sides:
c = (5/2)^2
Simplifying:
c = 25/4
Therefore, there exists a value c in the interval (4, 9) such that f'(c) = 1/5. One such possible value is c = 25/4.