Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
f(x) = √x [0, 9]
c=?
I've tried quite a few different answers
f(x) = √x
f(0) = 0
f(9) = 3
So, the slope of the secant is 1/3
f'(x) = 1 / 2√x
So, we want to find c such that
1 / 2√c = 1/3
√c = 3/2
c = 9/4
So, c is in the interval [0,9]
just to check,
f(9/4) = 3/2
So, the equation of the tangent line at c is
y = 1/3 (x-9/4) + 3/2
http://www.wolframalpha.com/input/?i=plot+y%3D%E2%88%9Ax%2C+y+%3D+1%2F3+%28x-9%2F4%29+%2B+3%2F2+for+x%3D0..9
To find the number c satisfying the conclusion of the Mean Value Theorem on the given interval [0, 9] for the function f(x) = √x, you need to verify two conditions:
1. f(x) must be continuous on the interval [0, 9].
2. f(x) must be differentiable on the open interval (0, 9).
Let's check if these conditions are met:
1. Continuity: The square root function, √x, is continuous on the interval [0, 9].
2. Differentiability: The square root function, √x, is differentiable on the open interval (0, 9).
Since both conditions are satisfied, we can use the Mean Value Theorem to find the value of c.
The Mean Value Theorem states that if f(x) is continuous on the interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in the open interval (a, b) such that:
f'(c) = (f(b) - f(a)) / (b - a)
So, to find the number c, we need to calculate f'(c) and check if it satisfies the equation.
Let's calculate the derivative of f(x):
f'(x) = (d/dx)(√x)
Using the power rule for differentiation, we get:
f'(x) = (1/2)x^(-1/2)
Now let's substitute the values for a, b, f(b), and f(a) into the Mean Value Theorem equation:
a = 0, b = 9, f(a) = √0 = 0, f(b) = √9 = 3
f'(c) = (3 - 0) / (9 - 0)
Simplifying, we have:
f'(c) = 1/3
Therefore, to find the number c, we need to solve the equation:
1/3 = (1/2)c^(-1/2)
Multiplying both sides by 2, we get:
2/3 = c^(-1/2)
Taking the reciprocal of both sides, we have:
3/2 = c^(1/2)
Squaring both sides, we find:
(3/2)^2 = c^(1/2)^2
9/4 = c
Therefore, the number c that satisfies the conclusion of the Mean Value Theorem on the interval [0, 9] for the function f(x) = √x is c = 9/4.