Let 𝑓(𝑥)=𝑥3−3𝑥2−9𝑥+25.Determine the absolute maximum and absolute minimum values of 𝑓 on the interval [0,4]

To find the absolute maximum and minimum values of a function on a closed interval [a, b], we need to evaluate the function at its critical points and endpoints, and compare the function values.

Step 1: Find the derivative of the function. f'(x) = 3x^2 - 6x - 9

Step 2: Set the derivative equal to zero to find the critical points. 3x^2 - 6x - 9 = 0 This simplifies to: x^2 - 2x - 3 = 0 Factoring gives: (x - 3)(x + 1) = 0 So, the critical points are x = 3 and x = -1.

Step 3: Evaluate the function at the critical points and endpoints. f(0) = 25 f(3) = 3^3 - 33^2 - 93 + 25 = -8 f(4) = 4^3 - 34^2 - 94 + 25 = -3 The critical point x = -1 is not in the interval [0, 4], so we ignore it.

Step 4: Compare the function values to find the absolute maximum and minimum. The absolute maximum value is 25 at x = 0, and the absolute minimum value is -8 at x = 3.