Question

Perform a first derivative test on the function f(x) = 2x⁵ - 10x⁴ + 10x³ +4; [-1,4]. a. Locate the critical points of the given function b. Use the First Derivative Test to locate the local maximum and minimum values. c. Identify the absolute maximum and minimum values of the function on the given interval

Answer 1

To perform a first derivative test on the function f(x) = 2x⁵ - 10x⁴ + 10x³ + 4 on the interval [-1,4], find the critical points by taking the derivative, use the first derivative test to locate local maximum and minimum values, and identify the absolute maximum and minimum values by evaluating the function at critical points and endpoints.

Step-by-step explanation:

To perform a first derivative test on the function f(x) = 2x⁵ - 10x⁴ + 10x³ + 4 on the interval [-1,4], follow these steps:

Find the critical points of the function by taking the derivative, setting it equal to zero, and solving for x. In this case, f'(x) = 10x⁴ - 40x³ + 30x². Setting this equal to zero, we get 10x²(x² - 4x + 3) = 0. The solutions are x = 0, x = 1, and x = 3.

Use the first derivative test to determine the local maximum and minimum values. Test the intervals (-∞,0), (0,1), (1,3), and (3,∞) by plugging in a test value into the first derivative. For example, plugging in x = -1 into f'(x) gives a positive value, indicating that the function is increasing on the interval (-∞,0). Plugging in x = 0.5 gives a negative value, indicating that the function is decreasing on the interval (0,1). Plugging in x = 2 gives a positive value, indicating that the function is increasing on the interval (1,3). Plugging in x = 4 gives a negative value, indicating that the function is decreasing on the interval (3,∞).

Identify the absolute maximum and minimum values of the function on the given interval. To do this, evaluate the function at the critical points and endpoints. In this case, f(-1) = 2(-1)⁵ - 10(-1)⁴ + 10(-1)³ + 4 = -2 + 10 - 10 + 4 = 2. f(4) = 2(4)⁵ - 10(4)⁴ + 10(4)³ + 4 = 128 - 640 + 640 + 4 = 132. The absolute maximum value is 132 at x = 4 and the absolute minimum value is 2 at x = -1.

Answer 2

The first derivative test involves finding the derivative of the function, setting it to zero to find critical points, and then using the test to determine local maxima and minima. To find absolute extrema, evaluate the function at critical points and interval endpoints.

Step-by-step explanation:

First Derivative Test for a Function

To perform a first derivative test on the function f(x) = 2x⁵ - 10x⁴ + 10x³ + 4 over the interval [-1,4], we would first find the critical points by taking the derivative of f(x) and setting it to zero or determining where the derivative does not exist.

Locating Critical Points

The first derivative of f(x) is f'(x) = 10x⁴ - 40x³ + 30x². By factoring, we can find the values of x where f'(x) equals zero. Setting the derivative equal to zero gives us potential critical points. We would then need to test each critical point using the First Derivative Test to see where the function f(x) changes from increasing to decreasing or vice versa, indicating local minima and maxima.

First Derivative Test

By the first derivative test, if the sign of f'(x) changes from positive to negative, we have a local maximum, and if it changes from negative to positive, we have a local minimum. We would substitute values from each interval around our critical points to determine the sign of f'(x).

Absolute Maximum and Minimum Values

To find the absolute maximum and minimum values on the interval [-1, 4], we would evaluate the function at the critical points and the endpoints of the interval. The largest value would be the absolute maximum and the smallest value would be the absolute minimum.