To find the minimum value of the function f(x) = 3x^4 - 8x^3 - 48x + 25 on the interval [0, 3], we need to find the critical points of the function and evaluate the function at these points and the endpoints of the interval.

Step 1: Find the derivative of the function.
f'(x) = 12x^3 - 24x^2 - 48

Step 2: Set the derivative equal to zero and solve for x to find the critical points.
12x^3 - 24x^2 - 48 = 0
Divide through by 12:
x^3 - 2x^2 - 4 = 0
This equation can be solved using methods such as the rational root theorem or synthetic division, or by using a graphing calculator or computer algebra system. The solutions are approximately x = -1.65063, x = 0.819172, and x = 2.83146. However, only x = 0.819172 and x = 2.83146 are in the interval [0, 3].

Step 3: Evaluate the function at the critical points and the endpoints of the interval.
f(0) = 3*0^4 - 8*0^3 - 48*0 + 25 = 25
f(0.819172) = 3*0.819172^4 - 8*0.819172^3 - 48*0.819172 + 25 ≈ -11.3838
f(2.83146) = 3*2.83146^4 - 8*2.83146^3 - 48*2.83146 + 25 ≈ 25.3838
f(3) = 3*3^4 - 8*3^3 - 48*3 + 25 = -10

Step 4: The minimum value of the function on the interval [0, 3] is the smallest of these values, which is approximately -11.3838.

Question

To find the minimum value of the function f(x) = 3x^4 - 8x^3 - 48x + 25 on the interval [0, 3], we need to find the critical points of the function and evaluate the function at these points and the endpoints of the interval.

Step 1: Find the derivative of the function.
f'(x) = 12x^3 - 24x^2 - 48

Step 2: Set the derivative equal to zero and solve for x to find the critical points.
12x^3 - 24x^2 - 48 = 0
Divide through by 12:
x^3 - 2x^2 - 4 = 0
This equation can be solved using methods such as the rational root theorem or synthetic division, or by using a graphing calculator or computer algebra system. The solutions are approximately x = -1.65063, x = 0.819172, and x = 2.83146. However, only x = 0.819172 and x = 2.83146 are in the interval [0, 3].

Step 3: Evaluate the function at the critical points and the endpoints of the interval.
f(0) = 3*0^4 - 8*0^3 - 48*0 + 25 = 25
f(0.819172) = 3*0.819172^4 - 8*0.819172^3 - 48*0.819172 + 25 ≈ -11.3838
f(2.83146) = 3*2.83146^4 - 8*2.83146^3 - 48*2.83146 + 25 ≈ 25.3838
f(3) = 3*3^4 - 8*3^3 - 48*3 + 25 = -10

Step 4: The minimum value of the function on the interval [0, 3] is the smallest of these values, which is approximately -11.3838.

Knowee AI · Accepted Answer

To find the minimum value of the function f(x) = 3x^4 - 8x^3 - 48x + 25 on the interval [0, 3], we need to find the critical points of the function and evaluate the function at these points and the endpoints of the interval.

Step 1: Find the derivative of the function.
f'(x) = 12x^3 - 24x^2 - 48

Step 2: Set the derivative equal to zero and solve for x to find the critical points.
12x^3 - 24x^2 - 48 = 0
Divide through by 12:
x^3 - 2x^2 - 4 = 0
This equation can be solved using methods such as the rational root theorem or synthetic division, or by using a graphing calculator or computer algebra system. The solutions are approximately x = -1.65063, x = 0.819172, and x = 2.83146. However, only x = 0.819172 and x = 2.83146 are in the interval [0, 3].

Step 3: Evaluate the function at the critical points and the endpoints of the interval.
f(0) = 3*0^4 - 8*0^3 - 48*0 + 25 = 25
f(0.819172) = 3*0.819172^4 - 8*0.819172^3 - 48*0.819172 + 25 ≈ -11.3838
f(2.83146) = 3*2.83146^4 - 8*2.83146^3 - 48*2.83146 + 25 ≈ 25.3838
f(3) = 3*3^4 - 8*3^3 - 48*3 + 25 = -10

Step 4: The minimum value of the function on the interval [0, 3] is the smallest of these values, which is approximately -11.3838.

The minimum value of f(x) = 3x4 - 8x3 - 48x + 25 on [0, 3] is

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