Question

Consider the following function on the given interval. f(x) = 18 4x − x2, [0, 5] find the derivative of the function. f ′(x) = correct: your answer is correct. find any critical numbers of the function. (enter your answers as a comma-separated list. if an answer does not exist, enter dne.) x = correct: your answer is correct. find the absolute maximum and absolute minimum values of f on the given interval. absolute minimum value incorrect: your answer is incorrect. absolute maximum value incorrect: your answer is incorrect.

Answer 1

To find the derivative of the function f(x) = 18 - 4x - x^2 on the interval [0, 5], we use the power rule and constant rule of differentiation. The critical number of the function is x = -2. The absolute maximum value is 18 and the absolute minimum value is -14.

Step-by-step explanation:

To find the derivative of the function f(x) = 18 - 4x - x^2 on the interval [0, 5], we can use the power rule and the constant rule of differentiation. The power rule states that the derivative of x^n is n*x^(n-1), and the constant rule states that the derivative of a constant is 0.

Applying the power rule to each term of the function, we have f'(x) = 0 - 4 - 2x = -4 - 2x.

To find the critical numbers of the function, we set the derivative equal to 0 and solve for x: -4 - 2x = 0 => x = -4/2 = -2. Therefore, the critical number of the function is x = -2.

Since the function f(x) is a quadratic with a negative coefficient for the x^2 term, it opens downwards. This means that it has a maximum value at the vertex. To find the absolute maximum and absolute minimum values of f on the interval [0, 5], we evaluate the function at the endpoints and the critical point. f(0) = 18, f(5) = -7, and f(-2) = -14. Therefore, the absolute maximum value is 18 and the absolute minimum value is -14.