Answer:
c = 9/4 = 2.25
Explanation:
You want the value of c that satisfies the Mean Value Theorem for f(x) = √x on the interval [1, 4].
Mean Value Theorem
The Mean Value Theorem (MVT) tells you that for some value x=c in the interval [a, b], the slope of continuous function f(x) will be equal to the average slope between (a, f(a)) and (b, f(b)), the function values at the end points of the interval.
Average slope
The end points of the interval are ...
(1, f(1)) = (1, 1)
(4, f(4)) = (4, 2)
Then the average slope is ...
m = (y2 -y1)/(x2 -x1) = (2 -1)/(4 -1) = 1/3 . . . . average rate of change
Derivative
The derivative of f(x) is ...
f'(x) = 1/2x^(-1/2) = 1/(2√x)
We want to find c such that f'(c) = 1/3:
1/3 = 1/(2√x)
√x = 3/2 . . . . . . . . multiply by 3√x
x = 9/4 . . . . . . . . square both sides
The value of c is 9/4 = 2.25.