Question

Prove that if $f^{\prime}(x)=g^{\prime}(x)$ for all $x$ in $(a, b),$ then there is a constant $C$ such that $f(x)=g(x)+C$ on $(a, b) .$ [Hint: Apply the Constant Function Theorem to $h(x)=f(x)-g(x) .]$

Name: Problem 26, Chapter 3: Calculus Single Variable
Uploaded: 2021-05-16T00:23:03-08:00
Duration: 1 min 3 s
Channel: Dharmendra Jain
Description: Prove that if f^'(x)=g^'(x) for all x in (a, b), then there is a constant C such that f(x)=g(x)+C on (a, b) . [Hint: Apply the Constant Function Theorem to h(x)=f(x)-g(x) .]

   Prove that if $f^{\prime}(x)=g^{\prime}(x)$ for all $x$ in $(a, b),$ then there is a constant $C$ such that $f(x)=g(x)+C$ on $(a, b) .$ [Hint: Apply the Constant Function Theorem to $h(x)=f(x)-g(x) .]$

Calculus Single Variable

Deborah… 7th Edition

Chapter 3, Problem 26

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Solved by Expert Dharmendra Jain

05/16/2021

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e., $h(x) = f(x) - g(x)$. Show more…

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Prove that if $f^{\prime}(x)=g^{\prime}(x)$ for all $x$ in $(a, b),$ then there is a constant $C$ such that $f(x)=g(x)+C$ on $(a, b) .$ [Hint: Apply the Constant Function Theorem to $h(x)=f(x)-g(x) .]$

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Key Concepts

Derivative

The derivative of a function represents its instantaneous rate of change. When two functions have equal derivatives over an interval, it means that they change at the same rate within that interval. This concept is fundamental in understanding how the behaviors of different functions can be compared by examining their rates of change.

Constant Function Theorem

The Constant Function Theorem states that if a function’s derivative is zero throughout an interval, then the function must be constant on that interval. This result is often derived from the Mean Value Theorem and is used to conclude that the difference of two functions with identical derivatives is a constant, showing that they differ only by a fixed value.

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