To find the absolute minimum and maximum values of the function [tex]\( f(x) = x^4 - 18x^2 + 9 \)[/tex] on the interval [tex]\([-2, 7]\)[/tex], follow these steps:
1. Define the function:
[tex]\[ f(x) = x^4 - 18x^2 + 9 \][/tex]
2. Find the critical points of the function:
- To find the critical points, we first need to calculate the derivative of the function:
[tex]\[ f'(x) = \frac{d}{dx}(x^4 - 18x^2 + 9) = 4x^3 - 36x \][/tex]
- Set the derivative equal to zero to find the critical points:
[tex]\[ 4x^3 - 36x = 0 \][/tex]
Factor out the common term:
[tex]\[ 4x(x^2 - 9) = 0 \][/tex]
Which simplifies to:
[tex]\[ 4x(x - 3)(x + 3) = 0 \][/tex]
This gives us the critical points [tex]\( x = 0, x = 3, x = -3 \)[/tex].
3. Evaluate the function at the critical points and the boundary points:
- The boundary points are [tex]\( x = -2 \)[/tex] and [tex]\( x = 7 \)[/tex].
- Calculate [tex]\( f(x) \)[/tex] at these points:
[tex]\[ f(-2) = (-2)^4 - 18(-2)^2 + 9 = 16 - 72 + 9 = -47 \][/tex]
[tex]\[ f(7) = 7^4 - 18(7^2) + 9 = 2401 - 882 + 9 = 1528 \][/tex]
- Calculate [tex]\( f(x) \)[/tex] at the critical points within the interval [tex]\([-2, 7]\)[/tex]:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 0^4 - 18(0^2) + 9 = 9 \][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = 3^4 - 18(3^2) + 9 = 81 - 162 + 9 = -72 \][/tex]
- [tex]\( x = -3 \)[/tex] is not within the interval [tex]\([-2, 7]\)[/tex], so we do not consider it.
4. Determine the absolute minimum and maximum values:
- We must compare the function values at the critical points and boundaries:
[tex]\[ f(-2) = -47 \][/tex]
[tex]\[ f(7) = 1528 \][/tex]
[tex]\[ f(0) = 9 \][/tex]
[tex]\[ f(3) = -72 \][/tex]
- From these calculations, we observe:
- The absolute minimum value is [tex]\( -72 \)[/tex] at [tex]\( x = 3 \)[/tex].
- The absolute maximum value is [tex]\( 1528 \)[/tex] at [tex]\( x = 7 \)[/tex].
Therefore, the absolute minimum value of [tex]\( f(x) \)[/tex] on the interval [tex]\([-2, 7]\)[/tex] is [tex]\(-72\)[/tex] and the absolute maximum value is [tex]\( 1528 \)[/tex].
