in exercises 17 - 26, find the extreme values of the function on the given interval. 18. f(x)=x³ - 9/2x²…

in exercises 17 - 26, find the extreme values of the function on the given interval. 18. f(x)=x³ - 9/2x² - 30x + 3 on 0,6.

in exercises 17 - 26, find the extreme values of the function on the given interval. 18. f(x)=x³ - 9/2x² - 30x + 3 on 0,6.

Answer

Answer:

The maximum value is (3) at (x = 0), and the minimum value is (-177) at (x = 5).

Explanation:

Step1: Find the derivative

Differentiate (f(x)=x^{3}-\frac{9}{2}x^{2}-30x + 3) using power - rule. (f'(x)=3x^{2}-9x - 30).

Step2: Set the derivative equal to zero

(3x^{2}-9x - 30 = 0). Divide through by (3): (x^{2}-3x - 10=0). Factor: ((x - 5)(x+2)=0). Solutions are (x = 5) and (x=-2). But (x=-2) is outside the interval ([0,6]), so we discard it.

Step3: Evaluate the function at critical and end - points

Evaluate (f(x)) at (x = 0), (x = 5), and (x = 6). (f(0)=0^{3}-\frac{9}{2}(0)^{2}-30(0)+3 = 3). (f(5)=5^{3}-\frac{9}{2}(5)^{2}-30(5)+3=125-\frac{225}{2}-150 + 3=125-112.5-150 + 3=-177). (f(6)=6^{3}-\frac{9}{2}(6)^{2}-30(6)+3=216 - 162-180 + 3=-123).