Consider the function f(x) = x^4 - 18 x^2 + 4, \quad
-2 \leq x \leq 7. This function has an absolute minimum value equal to
and an absolute maximum value equal to
Absolute minimum value: -64
Absolute maximum value: 476
To find the absolute minimum and maximum values of the function f(x) = x^4 - 18x^2 + 4 on the interval -2 ≤ x ≤ 7, we can follow these steps:
1. Find the critical points of the function by taking the derivative and setting it equal to zero.
2. Evaluate the function at the critical points and endpoints to determine the minimum and maximum values.
Step 1: Finding the critical points
To find critical points, we need to take the derivative of f(x) and set it equal to zero.
f'(x) = 4x^3 - 36x = 0
Factoring out a common factor, we have:
4x(x^2 - 9) = 0
This equation is equal to zero when either 4x = 0 or x^2 - 9 = 0.
So the critical points are x = 0, x = -3, and x = 3.
Step 2: Evaluate the function at the critical points and endpoints
Now, we need to evaluate the function f(x) at the critical points and endpoints (-2 and 7) to find the minimum and maximum values.
f(-2) = (-2)^4 - 18(-2)^2 + 4 = 16 - 72 + 4 = -52
f(0) = 0^4 - 18(0)^2 + 4 = 4
f(3) = 3^4 - 18(3)^2 + 4 = 81 - 162 + 4 = -77
f(7) = 7^4 - 18(7)^2 + 4 = 2401 - 882 + 4 = 1523
Now, we compare these values to determine the absolute minimum and maximum.
The absolute minimum value is -77, which occurs at x = 3.
The absolute maximum value is 1523, which occurs at x = 7.
So, the absolute minimum value is -77, and the absolute maximum value is 1523 for the given function on the interval -2 ≤ x ≤ 7.
To find the absolute minimum and maximum values of the given function, f(x) = x^4 - 18x^2 + 4 on the interval -2 ≤ x ≤ 7, we will follow these steps:
1. Find the critical points by taking the derivative of the function and setting it equal to zero:
f'(x) = 4x^3 - 36x = 0
Factoring out 4x:
4x(x^2 - 9) = 0
Setting each factor equal to zero:
4x = 0 ---> x = 0
x^2 - 9 = 0 ---> (x + 3)(x - 3) = 0 ---> x = -3, 3
So, the critical points are x = 0, x = -3, and x = 3.
2. Evaluate the function at the critical points and at the endpoints of the interval:
f(-2) = (-2)^4 - 18(-2)^2 + 4 = 16 - 72 + 4 = -52
f(0) = 0^4 - 18(0)^2 + 4 = 4
f(7) = 7^4 - 18(7)^2 + 4 = 2401 - 882 + 4 = 1523
f(-3) = (-3)^4 - 18(-3)^2 + 4 = 81 - 162 + 4 = -77
f(3) = 3^4 - 18(3)^2 + 4 = 81 - 162 + 4 = -77
3. Compare the values obtained in step 2 to determine the absolute minimum and maximum:
The smallest value is -77, which occurs at x = -3 and x = 3, so the absolute minimum value is -77.
The largest value is 1523, which occurs at x = 7, so the absolute maximum value is 1523.
Therefore, the absolute minimum value of the function f(x) = x^4 - 18x^2 + 4 on the interval -2 ≤ x ≤ 7 is -77, and the absolute maximum value is 1523.