Find the intervals in which the following function  f(x)=20−9x+6x2−x3𝑓𝑥=20−9𝑥+6𝑥2−𝑥3 is(a)𝑎 Strictly increasing,(b)𝑏 Strictly decreasing.

To find the intervals where the function is strictly increasing or decreasing, we first need to find its derivative. The derivative of a function gives us the slope of the function at any point, and the sign of the derivative tells us whether the function is increasing or decreasing.

The function is f(x) = 20 - 9x + 6x^2 - x^3.

Step 1: Find the derivative of the function. f'(x) = -9 + 12x - 3x^2.

Step 2: Set the derivative equal to zero and solve for x to find critical points. -9 + 12x - 3x^2 = 0 Rearranging, we get 3x^2 - 12x + 9 = 0 Dividing through by 3, we get x^2 - 4x + 3 = 0 Factoring, we get (x - 1)(x - 3) = 0 Setting each factor equal to zero gives us x = 1 and x = 3.

Step 3: Test the intervals determined by the critical points in the derivative to see where the function is increasing or decreasing. The critical points divide the x-axis into three intervals: (-∞, 1), (1, 3), and (3, ∞).

For the interval (-∞, 1), choose a test point, say x = 0. Substituting x = 0 into the derivative gives -9, which is negative. Therefore, the function is decreasing on the interval (-∞, 1).

For the interval (1, 3), choose a test point, say x = 2. Substituting x = 2 into the derivative gives 3, which is positive. Therefore, the function is increasing on the interval (1, 3).

For the interval (3, ∞), choose a test point, say x = 4. Substituting x = 4 into the derivative gives -3, which is negative. Therefore, the function is decreasing on the interval (3, ∞).

So, the function is strictly increasing on the interval (1, 3) and strictly decreasing on the intervals (-∞, 1) and (3, ∞).