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

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 the critical point 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., \( x = -1 \)): \[ G'(-1) = \frac{32(-1)}{((-1)^2 + 2)^2} = \frac{-32}{(1 + 2)^2} = \frac{-32}{9} < 0 \] (decreasing) - For \( x > 0 \) (e.g., \( x = 1 \)): \[ G'(1) = \frac{32(1)}{(1^2 + 2)^2} = \frac{32}{(1 + 2)^2} = \frac{32}{9} > 0 \] (increasing) Since \( G'(x) \) changes from negative to positive at \( x = 0 \), we conclude that there is a relative minimum at \( x = 0 \). ### Step 4: Find the value of the function at the critical point Now we find \( G(0) \): \[ G(0) = \frac{-16}{0^2 + 2} = \frac{-16}{2} = -8 \] ### Summary of the relative extreme point The function has a relative minimum at the point \( (0, -8) \). ### Step 5: Sketch the graph To sketch the graph of \( G(x) \): 1. The function \( G(x) \) approaches 0 as \( x \) approaches \( \pm \infty \). 2. The function is always negative since \( G(x) < 0 \) for all \( x \). 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 \( x \) goes to \( \pm \infty \) and has a minimum point at \( (0, -8) \). ### Graph Here is a rough sketch of the graph: ``` y | | * | * * | * * | * * | * * | * * | * * | * * | * * | * * |* * +---------------------- x ``` The minimum point is at \( (0, -8) \), and the graph approaches the x-axis as \( x \) moves away from the origin.