Q. 1 - Calculus

Question

Fill in the blanks to complete each of the following theorem statements: 

If $F$ and $G$ are _____ functions, then $F\text{'}\left(x\right)=G\text{'}\left(x\right)$ if and only if $G\left(x\right)$ = _____.

Step-by-Step Solution

Answer

If F and G are differentiable functions, then F'(x)=G'(x ) if and only if $G\left(x\right)=F\left(x\right)+c$

Step 1. Given Information

A theorem statement is given with blanks we have to fill the blanks with correct answers.

The given statement with blanks is:

If F and G are ____ functions, then F'(x)=G'(x ) if and only if G(x) = _____.

Step 2. Theorem statement

Functions with the Same Derivative Differ by a Constant Theorem:

If F and G are differentiable functions, then $F\text{'}\left(x\right)=G\text{'}\left(x\right)$ if and only if $F\left(x\right)=G\left(x\right)+c$ for some constant $c$.

Therefore the blanks are Differentiable and $G\left(x\right)=F\left(x\right)+c$.