To find the critical numbers of the function f(x) = (3/5)x^5 - 9x^3 + 2, we need to find the values of x where the derivative of the function is either zero or undefined. Let's calculate the derivative first:
f'(x) = 15/5x^4 - 27x^2 = 3x^4 - 27x^2
a) To find the critical numbers (x-coordinates), we need to solve the equation 3x^4 - 27x^2 = 0 for x:
3x^2(x^2 - 9) = 0
This equation factors as 3x^2(x + 3)(x - 3) = 0. Setting each factor to zero gives us three critical numbers: x = 0, x = -3, and x = 3.
b) To find the intervals on which the graph is increasing or decreasing, we can use the critical numbers and test points within each interval. However, since the interval is already specified as [-4, 5], we can determine the intervals based on the critical numbers and endpoints.
The critical numbers divide the interval [-4, 5] into four smaller intervals: [-4, -3], [-3, 0], [0, 3], and [3, 5].
We can determine if the function is increasing or decreasing within each interval by evaluating the derivative (f'(x)) at a test point in each interval:
For the interval [-4, -3], let's evaluate f'(-4) = 3(-4)^4 - 27(-4)^2 = 768, which is positive. Therefore, the function is increasing in this interval.
For the interval [-3, 0], let's evaluate f'(-3) = 3(-3)^4 - 27(-3)^2 = 0, which is neither positive nor negative. We need to test another point in this interval, such as f'(-2) = 3(-2)^4 - 27(-2)^2 = -96, which is negative. Therefore, the function is decreasing in this interval.
For the interval [0, 3], let's evaluate f'(0) = 3(0)^4 - 27(0)^2 = 0, which is neither positive nor negative. We need to test another point in this interval, such as f'(1) = 3(1)^4 - 27(1)^2 = -24, which is negative. Therefore, the function is decreasing in this interval.
For the interval [3, 5], let's evaluate f'(3) = 3(3)^4 - 27(3)^2 = 0, which is neither positive nor negative. We need to test another point in this interval, such as f'(4) = 3(4)^4 - 27(4)^2 = 768, which is positive. Therefore, the function is increasing in this interval.
Based on these calculations, the intervals on which the graph of the function f(x) = (3/5)x^5 - 9x^3 + 2 is increasing are [-4, -3] and [3, 5]. The intervals on which the graph is decreasing are [-3, 0] and [0, 3].
Please note that this analysis gives us information about the increasing and decreasing behavior of the function, but it doesn't provide specific local or absolute maximum and minimum points.