VeeEight said:(1) is Rolle's Theorem
(2) is the Mean Value Theorem, which is a generalization of Rolle's Theorem
I did not know (3) but that is interesting
(4) is correct, f(x) can be a constant function not equal to 0.
(5) is correct - take a(x) = f(x) - g(x). It's derivative is 0, so a(x) is constant (you can prove this using MVT).
VeeEight said:Oh sorry, I missed that
(1) states that f is defined on [a,b] but it is not necessarily continuous there. It is continuous on (a,b) (since it is differentiable there) but not necessarily at a or b.
The Mean Value Theorem is a fundamental theorem in calculus that states that for any continuous function on a closed interval, there exists at least one point in the interval where the instantaneous rate of change (derivative) is equal to the average rate of change. In other words, it guarantees the existence of a tangent line that is parallel to the secant line connecting the endpoints of the interval.
The Mean Value Theorem is used to prove other important theorems and formulas in calculus, such as the First and Second Derivative Tests for finding extrema, the Fundamental Theorem of Calculus for evaluating definite integrals, and the Intermediate Value Theorem for proving the existence of roots of equations.
The Mean Value Theorem requires that the function is continuous on a closed interval and differentiable on the open interval. This means that there cannot be any breaks or holes in the graph of the function, and the function must have a well-defined derivative at every point in the interval.
No, the Mean Value Theorem is only applicable to continuous functions on a closed interval. If a function is discontinuous or undefined at certain points, it cannot satisfy the conditions for the Mean Value Theorem to hold.
Rolle's Theorem is a special case of the Mean Value Theorem where the average and instantaneous rates of change are equal at the endpoints, resulting in a derivative of zero at some point in the interval. In other words, the Mean Value Theorem is a generalization of Rolle's Theorem, and both are used to prove important results in calculus.