Part 3/6:
Setting this equal to the average slope:
[
3C^2 - 1 = 3
]
Solving for ( C ) yields:
[
3C^2 = 4 \implies C^2 = \frac{4}{3} \implies C = \frac{2}{\sqrt{3}} \approx 1.15
]
This point ( C ) lies within the specified interval, confirming the existence of at least one point where the instantaneous rate of change matches the average rate of change.
Visual Interpretation
To visualize this relationship, we can graph the function ( f(x) = x^3 - x ) and the line representing the average slope ( y = 3x - 0 ), since at ( x = 0 ), ( f(0) = 0 ).