For a function to violate Rolle's theorem, it would need to do two things:NanaToru said:Homework Statement
Let f(x) = 1 - x2/3. Show that f(-1) = f(1) but there is no number c in (-1,1) such that f'(c) = 0. Why does this not contradict Rolle's Theorem?
Homework Equations
The Attempt at a Solution
f(x) = 1 - x2/3.
f(-1) = 1 - 1 = 0
f(1) = 1 - 1 = 0
f' = 2/3 x -1/3.
I don't understand why this doesn't have a number c in f'(c), or why Rolle's theorem excludes nondifferentiable points?
If you think ##\frac 1 0 = 0## then you must also think ##0\cdot 0 = 1##?NanaToru said:Is it not 0? Or is that not a valid answer?
Just wait!NanaToru said:...This is honestly the most embarrassing moment of my life. Thank you though!
The Mean Value Theorem is a fundamental theorem in calculus that states that for any continuous and differentiable function on an interval, there exists at least one point where the slope of the tangent line is equal to the average rate of change of the function over that interval.
Rolle's Theorem is a special case of the Mean Value Theorem where the function has the same value at the endpoints of the interval. It states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point where the derivative is equal to zero.
Rolle's Theorem is a special case of the Mean Value Theorem. Both theorems are used to prove the existence of a point where the derivative is equal to a certain value. However, Rolle's Theorem requires the function to have the same value at the endpoints of the interval, while the Mean Value Theorem only requires the function to be continuous and differentiable on the interval.
Differentiability is a key requirement in both Mean Value Theorem and Rolle's Theorem. These theorems rely on the existence of the derivative of a function at a certain point, and without differentiability, the theorems cannot be applied. In other words, the functions must be smooth and have a well-defined tangent line at the point in question.
Mean Value Theorem and Rolle's Theorem are used to solve a variety of problems in fields such as physics, engineering, and economics. For example, they can be used to analyze the movement of objects, optimize production processes, and determine the maximum or minimum values of a function. They also have applications in the study of optimization and optimization algorithms.