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To solve the problem of finding the absolute minimum and maximum values of the function [tex]\( f(x) = x^4 - 18x^2 + 9 \)[/tex] over the interval [tex]\([-2, 7]\)[/tex], we proceed with the following steps:
### Step 1: Evaluate endpoints of the interval
First, we evaluate the function at the endpoints of the given interval.
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = (-2)^4 - 18(-2)^2 + 9 = 16 - 72 + 9 = -47 \][/tex]
- For [tex]\( x = 7 \)[/tex]:
[tex]\[ f(7) = 7^4 - 18 \cdot 7^2 + 9 = 2401 - 882 + 9 = 1528 \][/tex]
These calculations help us know the function values at the boundaries.
### Step 2: Derivative and critical points
Next, we find the critical points by computing the derivative of [tex]\( f(x) \)[/tex] and setting it to zero.
The first derivative of [tex]\( f(x) \)[/tex] is:
[tex]\[ f'(x) = 4x^3 - 36x \][/tex]
To find the critical points, we solve [tex]\( f'(x) = 0 \)[/tex]:
[tex]\[ 4x^3 - 36x = 0 \][/tex]
[tex]\[ 4x(x^2 - 9) = 0 \][/tex]
[tex]\[ x(4)(x^2 - 9) = 0 \][/tex]
[tex]\[ x(x - 3)(x + 3) = 0 \][/tex]
Thus, the critical points are [tex]\( x = 0 \)[/tex], [tex]\( x = 3 \)[/tex], and [tex]\( x = -3 \)[/tex]. Note that -3 is not within our interval [tex]\([-2, 7]\)[/tex], so we will discard it.
### Step 3: Evaluate the function at critical points within the interval
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 0^4 - 18 \cdot 0^2 + 9 = 9 \][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = 3^4 - 18 \cdot 3^2 + 9 = 81 - 162 + 9 = -72 \][/tex]
### Step 4: Compare function values
Now, we compare all the function values that we have computed:
- Endpoint at [tex]\( x = -2 \)[/tex]: [tex]\( f(-2) = -47 \)[/tex]
- Endpoint at [tex]\( x = 7 \)[/tex]: [tex]\( f(7) = 1528 \)[/tex]
- Critical point at [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = 9 \)[/tex]
- Critical point at [tex]\( x = 3 \)[/tex]: [tex]\( f(3) = -72 \)[/tex]
From these values, we can identify the absolute minimum and maximum values in the interval:
- The absolute minimum value is [tex]\( f(3) = -72 \)[/tex] at [tex]\( x = 3 \)[/tex].
- The absolute maximum value is [tex]\( f(0) = 9 \)[/tex] at [tex]\( x = 0 \)[/tex].
Therefore, the absolute minimum value is [tex]\(-72\)[/tex] and the absolute maximum value is [tex]\(9\)[/tex].
### Step 1: Evaluate endpoints of the interval
First, we evaluate the function at the endpoints of the given interval.
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = (-2)^4 - 18(-2)^2 + 9 = 16 - 72 + 9 = -47 \][/tex]
- For [tex]\( x = 7 \)[/tex]:
[tex]\[ f(7) = 7^4 - 18 \cdot 7^2 + 9 = 2401 - 882 + 9 = 1528 \][/tex]
These calculations help us know the function values at the boundaries.
### Step 2: Derivative and critical points
Next, we find the critical points by computing the derivative of [tex]\( f(x) \)[/tex] and setting it to zero.
The first derivative of [tex]\( f(x) \)[/tex] is:
[tex]\[ f'(x) = 4x^3 - 36x \][/tex]
To find the critical points, we solve [tex]\( f'(x) = 0 \)[/tex]:
[tex]\[ 4x^3 - 36x = 0 \][/tex]
[tex]\[ 4x(x^2 - 9) = 0 \][/tex]
[tex]\[ x(4)(x^2 - 9) = 0 \][/tex]
[tex]\[ x(x - 3)(x + 3) = 0 \][/tex]
Thus, the critical points are [tex]\( x = 0 \)[/tex], [tex]\( x = 3 \)[/tex], and [tex]\( x = -3 \)[/tex]. Note that -3 is not within our interval [tex]\([-2, 7]\)[/tex], so we will discard it.
### Step 3: Evaluate the function at critical points within the interval
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 0^4 - 18 \cdot 0^2 + 9 = 9 \][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = 3^4 - 18 \cdot 3^2 + 9 = 81 - 162 + 9 = -72 \][/tex]
### Step 4: Compare function values
Now, we compare all the function values that we have computed:
- Endpoint at [tex]\( x = -2 \)[/tex]: [tex]\( f(-2) = -47 \)[/tex]
- Endpoint at [tex]\( x = 7 \)[/tex]: [tex]\( f(7) = 1528 \)[/tex]
- Critical point at [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = 9 \)[/tex]
- Critical point at [tex]\( x = 3 \)[/tex]: [tex]\( f(3) = -72 \)[/tex]
From these values, we can identify the absolute minimum and maximum values in the interval:
- The absolute minimum value is [tex]\( f(3) = -72 \)[/tex] at [tex]\( x = 3 \)[/tex].
- The absolute maximum value is [tex]\( f(0) = 9 \)[/tex] at [tex]\( x = 0 \)[/tex].
Therefore, the absolute minimum value is [tex]\(-72\)[/tex] and the absolute maximum value is [tex]\(9\)[/tex].
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