Suppose the average rate of change of $g\left(x\right)$ between $x=4$ and $x=7$ is $0$. Which statement must be true?

• A. $g\left(7\right)$=$g\left(4\right)$
• B. $g\left(7\right)$<$g\left(4\right)$
• C. $g\left(7\right)$>$g\left(4\right)$
• D. $g\left(x\right)$ is constant for all $x$

Answer: (A)

Verified step by step guidance

1. Step 1: Recall the formula for the average rate of change of a function g(x) between two points x = a and x = b. It is given by: $\frac{g\left(b\right)-g\left(a\right)}{b-a}$.
2. Step 2: Substitute the given values x = 4 and x = 7 into the formula. The average rate of change becomes: $\frac{g\left(7\right)-g\left(4\right)}{7-4}$.
3. Step 3: The problem states that the average rate of change is 0. Set the formula equal to 0: $\frac{g\left(7\right)-g\left(4\right)}{7-4}$ = 0.
4. Step 4: Multiply through by the denominator (7 - 4 = 3) to eliminate the fraction. This gives: $g\left(7\right)-g\left(4\right)=0$.
5. Step 5: Solve for g(7) and g(4). Rearrange the equation to find: $g\left(7\right)=g\left(4\right)$. This means the function g(x) has the same value at x = 4 and x = 7, which implies g(x) is constant between these points.