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Chapter 10: Problem 3

Zeigen Sie, dass eine differenzierbare Funktion \(f:(a, b) \rightarrow \mathbb{R}\) affin-linear ist, wenn ihre Ableitung konstant ist.

Short Answer

Expert verified
Question: Show that a differentiable function \(f:(a, b) \rightarrow \mathbb{R}\) is affine-linear if its derivative is constant. Answer: After finding that the derivative of the function is constant, we integrate the constant derivative to show that the original function is in the affine-linear form, \(f(x) = mx + c\), concluding that the function is affine-linear if its derivative is constant.

Step by step solution

01

Recall the definition of an affine-linear function

An affine-linear function is a function of the form \(f(x) = mx + c\), where \(m\) and \(c\) are constants. This means that the function has a constant slope and a constant vertical intercept. Our goal is to show that if a differentiable function has a constant derivative, it must be of this form.
02

Find the derivative of the affine-linear function and its properties

Since we know that \(f\) is differentiable, we can find its derivative \(f'(x)\). This is given by \(f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\). We are given that \(f'(x)\) is constant. This means \(f'(x)=m\) for all \(x\) in \((a, b)\) as \(m\) is a constant.
03

Integrate the constant derivative to obtain the original function

Since \(f'(x)\) is the derivative of \(f(x)\), we can find the function \(f\) by integrating \(f'(x)\): \(\int f'(x) dx = \int m dx\) Here, we obtain \(f(x) = mx + c\), where \(c\) is the integration constant. The integration constant represents the vertical intercept of the function. Therefore, \(f(x)\) is an affine-linear function with constant slope \(m\) and constant vertical intercept \(c\).
04

Conclusion

We have proven that a differentiable function \(f:(a, b) \rightarrow \mathbb{R}\) is affine-linear if its derivative is constant, as its form will always be \(f(x) = mx + c\), with constant slope \(m\) and constant vertical intercept \(c\).

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Most popular questions from this chapter

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