Find the relative extreme points of the function, if they exist. Then sketch a graph of the function. \( G(x)=\frac{-16}{x^{2}+2} \)

To find the relative extreme points of the function \( G(x) = \frac{-16}{x^2 + 2} \), we first need to find the derivative of the function and then determine where it is equal to zero or undefined. ### Step 1: Find the derivative Using the quotient rule, we can differentiate \( G(x) \): \[ G'(x) = \frac{(0)(x^2 + 2) - (-16)(2x)}{(x^2 + 2)^2} = \frac{32x}{(x^2 + 2)^2} \] ### Step 2: Set the derivative equal to zero To find critical points, we set the derivative equal to zero: \[ 32x = 0 \] This gives us: \[ x = 0 \] ### Step 3: Determine if it is a relative extreme point Next, we need to check the second derivative or use the first derivative test to determine if this critical point is a maximum or minimum. #### First Derivative Test We can evaluate the sign of \( G'(x) \) around \( x = 0 \): - For \( x < 0 \) (e.g., \( x = -1 \)): \[ G'(-1) = \frac{32(-1)}{((-1)^2 + 2)^2} = \frac{-32}{(1 + 2)^2} = \frac{-32}{9} < 0 \] - For \( x > 0 \) (e.g., \( x = 1 \)): \[ G'(1) = \frac{32(1)}{(1^2 + 2)^2} = \frac{32}{(1 + 2)^2} = \frac{32}{9} > 0 \] Since \( G'(x) \) changes from negative to positive at \( x = 0 \), this indicates that \( x = 0 \) is a relative minimum. ### Step 4: Find the value of \( G(0) \) Now we can find the value of the function at the critical point: \[ G(0) = \frac{-16}{0^2 + 2} = \frac{-16}{2} = -8 \] ### Summary of Relative Extreme Points The function \( G(x) \) has a relative minimum at the point \( (0, -8) \). ### Step 5: Sketch the graph To sketch the graph of \( G(x) = \frac{-16}{x^2 + 2} \): 1. The function is always negative since the numerator is negative and the denominator is always positive. 2. As \( x \) approaches \( \pm \infty \), \( G(x) \) approaches \( 0 \) from below. 3. The relative minimum occurs at \( (0, -8) \). The graph will look like a downward-opening curve that approaches the x-axis (y = 0) as \( x \) moves away from zero in both directions, with a minimum point at \( (0, -8) \). ### Graph Here is a rough sketch of the graph: ``` y | 0|------------------- (asymptote) | * -8|--------*---------*---------*---------*---------*--------- x | * | * | * | * | * | | * | * | * | * | * | | * | * | * | * | * | | * | * | * | * | * | | * | * | * | * | * | | * | * | * | * | * | | * | * | * | * | * | |* |* |* |* |* | ``` This sketch illustrates the behavior of the function, showing the relative minimum at \( (0, -8) \) and the asymptotic behavior as \( x \) approaches \( \pm \infty \).