Question

If $f^{(n)}(x)=0,$ what is $f ?$.

Name: Problem 55, Chapter 5: Calculus, Early Transcendentals
Uploaded: 2021-08-07T21:14:18-08:00
Duration: 1 min
Channel: Kian Manafi
Description: If f^(n)(x)=0, what is f ?.

   If $f^{(n)}(x)=0,$ what is $f ?$.

Calculus, Early Transcendentals

Dennis G. Zill,… 4th Edition

Chapter 5, Problem 55

Instant Answer

Solved by Expert Kian Manafi

08/07/2021

Step 1

This means that the nth derivative of the function $f(x)$ is equal to zero. Show more…

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If $f^{(n)}(x)=0,$ what is $f ?$.

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Key Concepts

nth Derivative

The nth derivative refers to the result of differentiating a function n times consecutively. In many areas of calculus, analyzing higher order derivatives helps determine the behavior and form of a function, especially when certain derivatives vanish.

Polynomial Function

A polynomial function is an expression involving sums of powers of a variable multiplied by constants. When the nth derivative of a function is zero, it implies that any term of degree n or higher is absent, so the function must be a polynomial of degree at most n–1.

Integration and Constant of Integration

Since differentiation and integration are inverse processes, if a derivative of a function is zero, integrating zero repeatedly generates new constants. This process shows that the original function can be built up as a polynomial, with each integration step contributing a constant term.

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