Answer :
To determine where the graph of \( f(x) \) is on or below the graph of \( g(x) \), we need to solve the inequality:
[tex]\[ f(x) \leq g(x) \][/tex]
Given:
[tex]\[ f(x) = x^4 - 72 \][/tex]
[tex]\[ g(x) = x^2 \][/tex]
So the inequality becomes:
[tex]\[ x^4 - 72 \leq x^2 \][/tex]
We can rewrite this inequality as:
[tex]\[ x^4 - x^2 - 72 \leq 0 \][/tex]
Let's solve this step by step:
1. Rearrange the terms to form a polynomial inequality:
[tex]\[ x^4 - x^2 - 72 \leq 0. \][/tex]
2. Introduce a substitution, let \( y = x^2 \):
[tex]\[ y^2 = x^4 \][/tex]
Thus the inequality becomes:
[tex]\[ y^2 - y - 72 \leq 0 \][/tex]
3. Solve the quadratic equation \( y^2 - y - 72 = 0 \) to find the critical points:
To solve \( y^2 - y - 72 = 0 \), we use the quadratic formula:
[tex]\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where \( a = 1 \), \( b = -1 \), and \( c = -72 \). Substituting these values in, we get:
[tex]\[ y = \frac{1 \pm \sqrt{1 + 288}}{2} \][/tex]
[tex]\[ y = \frac{1 \pm \sqrt{289}}{2} \][/tex]
[tex]\[ y = \frac{1 \pm 17}{2} \][/tex]
So:
[tex]\[ y = \frac{18}{2} = 9 \][/tex]
or
[tex]\[ y = \frac{-16}{2} = -8 \][/tex]
Since \( y = x^2 \) and \( x^2 \) must be non-negative, we discard \( y = -8 \) and keep \( y = 9 \).
4. Convert back to \( x \):
[tex]\[ x^2 = 9 \][/tex]
This leads to:
[tex]\[ x = \pm 3 \][/tex]
5. Determine the sign of the polynomial between the intervals:
We need to test the intervals defined by \( x = -3 \) and \( x = 3 \). The critical points divide the number line into three intervals: \( (-\infty, -3) \), \( (-3, 3) \), and \( (3, \infty) \).
- For \( x \) in \( (-\infty, -3) \):
Pick \( x = -4 \):
[tex]\[ (-4)^4 - (-4)^2 - 72 = 256 - 16 - 72 = 168 \][/tex] (positive)
- For \( x \) in \( (-3, 3) \):
Pick \( x = 0 \):
[tex]\[ 0^4 - 0^2 - 72 = -72 \][/tex] (negative)
- For \( x \) in \( (3, \infty) \):
Pick \( x = 4 \):
[tex]\[ (4)^4 - (4)^2 - 72 = 256 - 16 - 72 = 168 \][/tex] (positive)
6. Write the solution in interval notation:
The inequality \( x^4 - x^2 - 72 \leq 0 \) is satisfied in the interval where the polynomial is negative or zero. From our analysis, this is when:
[tex]\[ -3 \leq x \leq 3 \][/tex]
Thus, the answer is:
[tex]\[ [-3, 3] \][/tex]
Therefore, the graph of \( f \) is on or below the graph of \( g \) over the interval \( [-3, 3] \).
Now, let's look at the graph options and choose the one where [tex]\( f(x) \)[/tex] is on or below [tex]\( g(x) \)[/tex] over the interval [tex]\( [-3, 3] \)[/tex]. Typically, such plots would show two curves [tex]\( f \)[/tex] and [tex]\( g \)[/tex] and the correct graph would display the function [tex]\( f(x) = x^4 - 72 \)[/tex] dipping below and touching the function [tex]\( g(x) = x^2 \)[/tex] exactly at [tex]\( x = -3 \)[/tex] and [tex]\( x = 3 \)[/tex]. Choose the graph that correctly represents this behavior.
[tex]\[ f(x) \leq g(x) \][/tex]
Given:
[tex]\[ f(x) = x^4 - 72 \][/tex]
[tex]\[ g(x) = x^2 \][/tex]
So the inequality becomes:
[tex]\[ x^4 - 72 \leq x^2 \][/tex]
We can rewrite this inequality as:
[tex]\[ x^4 - x^2 - 72 \leq 0 \][/tex]
Let's solve this step by step:
1. Rearrange the terms to form a polynomial inequality:
[tex]\[ x^4 - x^2 - 72 \leq 0. \][/tex]
2. Introduce a substitution, let \( y = x^2 \):
[tex]\[ y^2 = x^4 \][/tex]
Thus the inequality becomes:
[tex]\[ y^2 - y - 72 \leq 0 \][/tex]
3. Solve the quadratic equation \( y^2 - y - 72 = 0 \) to find the critical points:
To solve \( y^2 - y - 72 = 0 \), we use the quadratic formula:
[tex]\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where \( a = 1 \), \( b = -1 \), and \( c = -72 \). Substituting these values in, we get:
[tex]\[ y = \frac{1 \pm \sqrt{1 + 288}}{2} \][/tex]
[tex]\[ y = \frac{1 \pm \sqrt{289}}{2} \][/tex]
[tex]\[ y = \frac{1 \pm 17}{2} \][/tex]
So:
[tex]\[ y = \frac{18}{2} = 9 \][/tex]
or
[tex]\[ y = \frac{-16}{2} = -8 \][/tex]
Since \( y = x^2 \) and \( x^2 \) must be non-negative, we discard \( y = -8 \) and keep \( y = 9 \).
4. Convert back to \( x \):
[tex]\[ x^2 = 9 \][/tex]
This leads to:
[tex]\[ x = \pm 3 \][/tex]
5. Determine the sign of the polynomial between the intervals:
We need to test the intervals defined by \( x = -3 \) and \( x = 3 \). The critical points divide the number line into three intervals: \( (-\infty, -3) \), \( (-3, 3) \), and \( (3, \infty) \).
- For \( x \) in \( (-\infty, -3) \):
Pick \( x = -4 \):
[tex]\[ (-4)^4 - (-4)^2 - 72 = 256 - 16 - 72 = 168 \][/tex] (positive)
- For \( x \) in \( (-3, 3) \):
Pick \( x = 0 \):
[tex]\[ 0^4 - 0^2 - 72 = -72 \][/tex] (negative)
- For \( x \) in \( (3, \infty) \):
Pick \( x = 4 \):
[tex]\[ (4)^4 - (4)^2 - 72 = 256 - 16 - 72 = 168 \][/tex] (positive)
6. Write the solution in interval notation:
The inequality \( x^4 - x^2 - 72 \leq 0 \) is satisfied in the interval where the polynomial is negative or zero. From our analysis, this is when:
[tex]\[ -3 \leq x \leq 3 \][/tex]
Thus, the answer is:
[tex]\[ [-3, 3] \][/tex]
Therefore, the graph of \( f \) is on or below the graph of \( g \) over the interval \( [-3, 3] \).
Now, let's look at the graph options and choose the one where [tex]\( f(x) \)[/tex] is on or below [tex]\( g(x) \)[/tex] over the interval [tex]\( [-3, 3] \)[/tex]. Typically, such plots would show two curves [tex]\( f \)[/tex] and [tex]\( g \)[/tex] and the correct graph would display the function [tex]\( f(x) = x^4 - 72 \)[/tex] dipping below and touching the function [tex]\( g(x) = x^2 \)[/tex] exactly at [tex]\( x = -3 \)[/tex] and [tex]\( x = 3 \)[/tex]. Choose the graph that correctly represents this behavior.
