Question

Let \(f\left( x \right) = \frac{1}{x}\) and

\(g\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{\frac{1}{x}}&{{\rm{if}}\,x > 0}\\{1 + \frac{1}{x}}&{{\rm{if}}\,x < 0}\end{array}} \right.\)

Show that \(f'\left( x \right) = g'\left( x \right)\) for all \(x\) in their domains. Can we conclude from Corollary 7 that \(f - g\) is constant?

Short answer

It is proved that Corollary 7 cannot be concluded using the given functions.

Step-by-step solution

Write the Corollary 7

If \(f'\left( x \right) = g'\left( x \right)\) for all \(x\) in an interval \(\left( {a,b} \right)\), then \(f - g\) is constant on \(\left( {a,b} \right)\); that is, \(f\left( x \right) = g\left( x \right) + c\) where \(c\) is a constant.

The derivative of the function

For all \(x > 0\):

It is given that \(f\left( x \right) = \frac{1}{x}\). So, \(f'\left( x \right) = - \frac{1}{{{x^2}}}\) for all \(x > 0\). Also \(g\left( x \right) = \frac{1}{x}\). So, \(g'\left( x \right) = - \frac{1}{{{x^2}}}\) for all \(x > 0\).

For all \(x < 0\):

It is given that \(f\left( x \right) = \frac{1}{x}\). So, \(f'\left( x \right) = - \frac{1}{{{x^2}}}\) for all \(x < 0\). Also \(g\left( x \right) = 1 + \frac{1}{x}\). So, \(g'\left( x \right) = - \frac{1}{{{x^2}}}\) for all \(x < 0\).

Since, \(f\left( x \right)\) and \(g\left( x \right)\) are not defined on \(x = 0\). Therefore, \(f'\left( x \right) = g'\left( x \right)\) in their domain.

Check for Corollary 7

It is proved that \(f'\left( x \right) = g'\left( x \right)\) in their domain. Now, find \(f - g\) in their domain.

For all \(x > 0\):

It is given that \(f\left( x \right) = \frac{1}{x}\) and \(g\left( x \right) = \frac{1}{x}\). So, \(f\left( x \right) - g\left( x \right) = \frac{1}{x} - \frac{1}{x} = 0\) for all \(x > 0\).

For all \(x < 0\):

It is given that \(f\left( x \right) = \frac{1}{x}\) and \(g\left( x \right) = 1 + \frac{1}{x}\). So, \(f\left( x \right) - g\left( x \right) = \frac{1}{x} - 1 - \frac{1}{x} = - 1\) for all \(x < 0\).

Therefore, \(f - g\) is not constant in their domain.

Hence, Corollary 7 cannot be concluded using the given functions.