Question

If \(f'\left( x \right) = c\)(c a constant) for all \(x\), use Corollary 7 to show that \(f\left( x \right) = cx + d\) for some constant \(d\).

Short answer

It is proved that \(f\left( x \right) = cx + d\) for some constant \(d\).

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.

Proof the statement

Given that \(f'\left( x \right) = c\) where \(c\) is a constant. Let a function \(g\left( x \right) = cx\). So, the derivative \(g'\left( x \right) = c\).

Now, \(f'\left( x \right) = g'\left( x \right) = c\). So, by corollary 7 it can be said that \(f\left( x \right) - g\left( x \right)\) is a constant. Let \(d\) be the constant.

So, \(f\left( x \right) - g\left( x \right) = d \Rightarrow f\left( x \right) = g\left( x \right) + d\).

Hence, it is proved that \(f\left( x \right) = g\left( x \right) + d\) using corollary 7.