Final answer:
To find the extrema of the function f(x) = 3x⁴ − 4x³ − 12x² + 5 on the interval [−2,3], we calculate its derivative, set it to zero, and find the critical points. Then we evaluate these critical points, including the interval's endpoints, to determine the minimum and maximum values of the function.
Step-by-step explanation:
To find the extreme values, or extrema, of the function f(x) = 3x⁴ − 4x³ − 12x² + 5 on the interval I = [−2,3], we first need to find the function's critical points within this interval. Critical points are found by determining where the derivative f'(x) is zero or undefined. The derivative of f(x) is f'(x) = 12x³ − 12x² − 24x. Setting this equal to zero and factoring, we find the critical points at x = 0, x = 1, and x = -2 (noting that -2 is at the boundary of our interval).
Next, we evaluate the function f(x) at the critical points and the endpoints of the interval to find the extreme values. f(-2) = 3(-2)⁴ − 4(-2)³ − 12(-2)² + 5, f(0) = 5, f(1) = 3(1)⁴ − 4(1)³ − 12(1)² + 5, and f(3) = 3(3)⁴ − 4(3)³ − 12(3)² + 5. After computing these values, the largest and smallest results will be the maximum and minimum values of f(x) on the given interval.