To find the absolute extrema for the function on the specified domain, we identify the critical points by taking the derivative, setting it equal to zero, and solving for x. Then we evaluate the function at these critical points and the domain endpoints to determine the absolute minimum and maximum values.
To find the absolute extrema of the function f(x)=x³-10x²+25x+4 on the domain [0,9], we first need to determine the critical points by taking the derivative of the function and setting it equal to zero. The critical points occur where f'(x)=0 or where the derivative does not exist. Since we are dealing with a polynomial function, the derivative will exist everywhere, so we only need to find where it equals zero.
The derivative of the function is f'(x) = 3x² - 20x + 25. Setting this equal to zero yields a quadratic equation: 3x² - 20x + 25 = 0. Using the quadratic formula x = (-b ± √(b² - 4ac))/(2a), we can find the values of x that make the derivative zero. After finding the critical points, we evaluate the original function at these points as well as at the endpoints of the domain, x = 0 and x = 9, to determine the absolute minimum and maximum values.
The options given correspond to potential values for the absolute minimum. We need to plug all critical points and endpoints back into the original function to check which yields the smallest function value. This process will reveal the absolute minimum.