Given the function #f(x)=(x-3)^(2/3)#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [1,4] and find the c?
1 Answer
Please see below.
Explanation:
The Mean Value Theorem has two hypotheses:
H1 :
H2 :
We say that we can apply the Mean Value Theorem if both hypotheses are true.
H1 : The function
Because power functions are continuous on their domains and linear functions are continuous everywhere. And the composition of continuous functions is continuous.
H2 : The function
Because the derivative,
A note on "if . . . then . . . " theorems
If the hypotheses are not true, we learn nothing about the truth of the conclusion.
To determine whether the conclusion is true or false we try to solve the equation in the conclusion of MVT.
Solving
Here is a graph of the function and the secant line through the points with
graph{(y-(x-3)^(2/3))((y-1)-(1-4^(1/3))/(3)(x-4))=0 [-15.1, 10.21, -5.03, 7.63]}