Let's check each of the given statements:
1. Over the interval [2, 3], the average rate of change of g is lower than that of both f and h.
The statement is FALSE. The average rate of change of g is HIGHER than f and h.
2. As x increases on the interval [0, ∞), the rate of change of f eventually exceeds the rate of change of both g and h.
The statement is TRUE. f will exceed the rate of change of both g and h exponentially.
3. When x > 4, the value of f(x) exceeds the values of both g(x) and h(x).
Let's try to determine the value at x = 5 on each function to see if it's true.
[tex]\text{ f(5) = }3^5\text{ + 2 = 245}[/tex][tex]\text{ g(5) = }20(5)\text{ + 4 = }100\text{ + 4 = 104}[/tex][tex]\text{ h(5) = }2(5)^2\text{ + }5(5)\text{ + }2\text{ = 50 + 25 + 2 = 77}[/tex]
Therefore, the statement is TRUE.
4. As x increases on the interval [0, ∞), the rate of change of g eventually exceeds the rate of change of both f and h.
The statement is FALSE. f will exceed the rate of change of both g and h exponentially.
5. When x ≈ 8, the value of h(x) exceeds the values of both f(x) and g(x).
The statement is FALSE. f will exceed the values of both g(x) and h(x) at x = 8.
6. A quantity increasing exponentially eventually exceeds a quantity growing quadratically or linearly.
The statement is TRUE. Proof on the statement 3 example.
Therefore, only Statements 2, 3 and 6 are TRUE.
