Answer:
Explanation:
To find the rate of increase for the function f(x), we need to find the derivative of the function with respect to x. Let's do this step by step:
1. Given function: f(x) = (³/24)² * x^3.
2. We need to find the derivative of f(x) with respect to x.
3. To find the derivative, we can apply the power rule, which states that if we have a function of the form f(x) = ax^n, where a and n are constants, the derivative is given by f'(x) = anx^(n-1).
4. Applying the power rule to our function, we get f'(x) = 3(³/24)² * 3x^(3-1).
5. Simplifying the expression, we have f'(x) = (³/8) * 3x^2.
6. Further simplifying, f'(x) = 9x^2/8.
7. The rate of increase is the value of the derivative at a specific point. In this case, the rate of increase is f'(x) evaluated at x = 3.
8. Substituting x = 3 into the derivative, we get f'(3) = 9(3)^2/8 = 81/8.
9. Therefore, the rate of increase for the function f(x) = (³/24)² * x^3 at x = 3 is 81/8.
In summary, the rate of increase for the function f(x) = (³/24)² * x^3 at x = 3 is 81/8.