Question

If \({f^\prime }(x) = c\) (c a constant) for all x, use Corollary 7 to show that \(f(x) = cx + d\) for some constant d.

Short answer

The given statement \(f(x) = cx + d\) for some constant \(d\) if \({f^\prime }(x) = c\) where \(c\) is constant for all \(x\)is proved.

Step-by-step solution

Given data

The function is \(f(x) = cx + d\).

Concept of corollary

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

Prove the statement \(f(x) = cx + d\) for some constant \(d\) 

From the given condition, it is observed that \({f^\prime }(x) = c\).

In order to use the corollary mentioned above, define the function \(g(x)\) to be \(g(x) = cx\) for all \(x\) and for a constant \(c\).

Differentiate this will yield \({g^\prime }(x) = c\) for all \(x\).

From the given condition and from the above equation, obtain \({f^\prime }(x)\) as follows:

\(\begin{aligned}{c}{f^\prime }(x) &= {g^\prime }(x)\\ &= c\end{aligned}\)

Use the corollary mentioned above as follows:

So, \(f(x) = g(x) + d\) for some constant \(d\).

Substitute the expression \(g(x) = cx\) in \(f(x)\) as follows:

\(f(x) = cx + d\)

Hence, the given statement is proved.