The question asks to determine which pair of functions exhibits the exponential function consistently growing at a faster rate than the quadratic function over the interval $0 \leq x \leq 5$.

Analysis of the Graph:

From the image:

• Blue Curve: Appears to be a quadratic function, as it has the characteristic parabolic shape.
• Green Curve: Appears to be an exponential function, as it exhibits rapid growth after an initial slow increase.

Now, focusing on the interval $0 \leq x \leq 5$, let's break it down:

• For $x \in [0, 1]$: It appears that the quadratic (blue curve) grows faster than the exponential (green curve) because the blue curve starts at higher values.
• For $x > 1$: The exponential curve begins to grow at a much faster rate, overtaking the quadratic.
• By the time $x$ reaches 5, the exponential function is growing at a much faster rate compared to the quadratic.

Conclusion:

In this case, the green curve (exponential) grows faster than the blue curve (quadratic) for values of $x$ greater than 1, and remains consistently faster for $x > 1$.

Let me know if you need more details or clarifications.

Related Questions:

1. How can we analytically compare the growth rates of quadratic and exponential functions?
2. Why does the exponential function overtake the quadratic in the long run?
3. What happens if we extend the interval beyond $x = 5$?
4. How does the initial condition (starting values) affect the comparison of these functions?
5. Could any transformations of the functions alter their behavior over this interval?

Tip:

Exponential functions will eventually outgrow any polynomial function for sufficiently large values of $x$, due to their inherent faster growth rate.