Chapter 3.1, Problem 36E

To determine

Whether the statement is true or false.

Answer to Problem 36E

The statement is true.

Explanation of Solution

Given information:

The statement: “ If $f\left(x\right)=x^2+x$ , then $f\text{'}\left(x\right)$ exist for every real number $x$ .”

Formula used:

Definition of the derivative:

  $\frac{d}{dx}f\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$

Where $f\left(x\right)$ is the function and $h$ is a constant.

Find the limit of the function, use the derivative formula $\frac{d}{dx}f\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$ .

  $\begin{array}{c}f\text{'}\left(x\right)=\underset{x\to 0^{+}}{\lim }\frac{\left[{\left(x+h\right)}^2+\left(x+h\right)\right]-\left[x^2+x\right]}{h}\\ =\underset{x\to 0^{+}}{\lim }\frac{x^2+2hx+h^2+x+h-x^2-x}{h}\\ =\underset{x\to 0^{+}}{\lim }\frac{2hx+h^2+h}{h}\\ =\underset{x\to 0^{+}}{\lim }\left[2x+h+1\right]\\ =2x+1\end{array}$

Since the function $f\text{'}\left(x\right)=2x+1$ represent a polynomial, thus the function represent a real value of $x$ .

Hence, the statement is true.