Find a formula for a constant function using the mean value theorem

• lamictal
In summary, the conversation discusses using the Mean Value Theorem to prove that f(x) = x^3 + c for some c ϵ R when given the condition f'(x) = 3x^2. The person attempted to use two functions with the same derivative and reached a false conclusion, but the correct approach is to work with the function f(x) - x^3.
lamictal

Homework Statement

Let x ϵ R such that f'(x) = 3x^2. Prove that f(x) = x^3 + c for some c ϵ R using the Mean Value Theorem.

The Attempt at a Solution

I used two functions f(x) and g(x) that have the same derivative namely f'(x). Applying the theorem I am able to work up until f(x) = g(x), I am unsure of where to go from there.

If you reached the conclusion $$f(x) = g(x)$$ from only the hypothesis $$f'(x) = g'(x)$$, something is clearly wrong, since that deduction is false.

You have a function that's expected to be a constant, namely $$f(x) - x^3$$. Work with that.

What is the mean value theorem?

The mean value theorem is a fundamental theorem in calculus that states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the interval where the slope of the tangent line is equal to the slope of the secant line connecting the endpoints of the interval.

What is a constant function?

A constant function is a type of function in which every input value is mapped to the same output value. It is represented by a horizontal line on a graph and has a constant slope of 0.

Why is the mean value theorem important for finding a formula for a constant function?

The mean value theorem provides a way to prove the existence of a point where the slope of the tangent line is equal to the slope of the secant line. This point is crucial in determining the constant value of the function, which can then be used to find the formula for the function.

How do you use the mean value theorem to find a formula for a constant function?

To find a formula for a constant function using the mean value theorem, you first need to identify the closed interval on which the function is continuous and differentiable. Next, you need to find the slope of the secant line connecting the endpoints of the interval. Finally, you can use the mean value theorem to find the point where the slope of the tangent line is equal to the slope of the secant line, which will give you the constant value of the function.

Can the mean value theorem be used to find a formula for any type of function?

Yes, the mean value theorem can be used to find a formula for any type of function that is continuous on a closed interval and differentiable on the open interval. However, it is most commonly used for finding the formula for constant functions, as they have a constant value that can be determined using the mean value theorem.

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