Chapter 4.8, Problem 24ES

(a) Use the Constant Difference Theorem (4.8.3) to show that if $f^{\prime }\left(x\right)=g^{\prime }\left(x\right)$ for all $x$ in $\left(-\infty ,+\infty \right),$ and if $f\left(x_0\right)-g\left(x_0\right)=c$ at some point $x_0,$ then $f\left(x\right)-g\left(x\right)=c$ for all $x$ in $\left(-\infty ,+\infty \right)$ .

(b) Use the result in part (a) to show that the function $h\left(x\right)={\left(x-1\right)}^3-\left(x^2+3\right)\left(x-3\right)$ is constant for all $x$ in $\left(-\infty ,+\infty \right),$ and find the constant.

(c) Check the result in part (b) by multiplying out and simplifying the formula for $h\left(x\right)$ .