The function f(x) = 5x sqrt x+2 satisfies the hypotheses of the Mean Value Theorem on the interval [0,2]. Find all values of c that satisfy the conclusion of the theorem.
How would you use the MVT? I tried taking the derivative, in which resulted in 5sqrtx+2 + (5x/2sqrtx+2) and by using the MVT, f'(c) = f(b)-f(a)/b-a, in which was f(2)-f(0)/2-0.
Now, I have to plug in the values...but I am not sure how you would do that and find the values of c. Please help?
f(x) = 5x√(x+2) is continuous and differentiable on [0,2], so the MVT applies.
f(2) = 20
f(0) = 0
So, the slope of the secant is 10
MVT says that f'(c) = 20 somewhere in [0,2]
f'(x) = 5(3x+4) / 2√(x+2)
f'(c) = 0 if
5(3c+4) / 2√(c+2) = 10
5(3c+4) = 20√(c+2)
(3c+4)^2 = 16(c+2)
9c^2 + 24c + 16 = 16c+32
9c^2 + 8c - 16 = 0
c = 0.96101
That is in the interval [0,2], as predicted by the MVT.
To use the Mean Value Theorem (MVT) in this problem, we need to first verify that the function f(x) satisfies the hypotheses of the MVT on the given interval [0,2]. These hypotheses are:
1. f(x) is continuous on the interval [0,2].
2. f(x) is differentiable on the open interval (0,2).
Let's check each hypothesis:
1. The function f(x) = 5x√(x+2) involves basic algebraic operations and the square root function, both of which are continuous on their domains. So, f(x) is continuous on the interval [0,2].
2. To check differentiability, we need to find the derivative of f(x). You correctly found the derivative to be f'(x) = 5√(x+2) + (5x/2√(x+2)). Notice that the derivative is defined for x > -2 since the square root function is not defined for negative numbers.
Now, we can use the MVT:
The MVT states that there exists at least one value c in the open interval (0,2) such that the derivative of f at c is equal to the average rate of change of f over the interval [0,2].
Average rate of change of f on [0,2] = (f(2) - f(0))/(2 - 0)
To find the values of c that satisfy the conclusion of the MVT, we need to find the exact values of f(2) and f(0) and evaluate the expression (f(2) - f(0))/(2 - 0).
f(2) = 5(2)√(2+2) = 20√4 = 40
f(0) = 5(0)√(0+2) = 0
Average rate of change of f on [0,2] = (f(2) - f(0))/(2 - 0) = 40/2 = 20
Now, we need to find the value of c in (0,2) such that f'(c) = 20.
To solve this equation, we set f'(x) = 20:
5√(x+2) + (5x/2√(x+2)) = 20
Now, you solve for x, and the solution(s) will give you the value(s) of c that satisfy the conclusion of the MVT.
I hope this helps you understand how to use the Mean Value Theorem and find the values of c! Let me know if you have any further questions.