\( y=\frac{6 x}{x^{2}+36} \) Identify the coordinates of the absolute maximum points. Select the correct

To find the absolute maximum points of the function \( y=\frac{6 x}{x^{2}+36} \), we need to find the critical points and then determine which of these points correspond to the absolute maximum. 1. Find the critical points: To find the critical points, we need to find the values of \( x \) where the derivative of the function is equal to zero or undefined. 2. Find the derivative of the function: The derivative of the function \( y=\frac{6 x}{x^{2}+36} \) with respect to \( x \) is given by: \[ \frac{dy}{dx} = \frac{(x^2+36)(6) - 6x(2x)}{(x^2+36)^2} \] 3. Set the derivative equal to zero and solve for \( x \): \[ \frac{(x^2+36)(6) - 6x(2x)}{(x^2+36)^2} = 0 \] Solving this equation will give us the critical points. 4. Determine the absolute maximum points: After finding the critical points, we need to determine which of these points correspond to the absolute maximum. This can be done by evaluating the function at each critical point and comparing the values. Let's proceed with finding the critical points. Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{\left(x^{2}+36\right)\times 6-6x\times 2x}{\left(x^{2}+36\right)^{2}}=0\) - step1: Simplify: \(\frac{-6x^{2}+216}{\left(x^{2}+36\right)^{2}}=0\) - step2: Cross multiply: \(-6x^{2}+216=\left(x^{2}+36\right)^{2}\times 0\) - step3: Simplify the equation: \(-6x^{2}+216=0\) - step4: Move the constant to the right side: \(-6x^{2}=-216\) - step5: Change the signs: \(6x^{2}=216\) - step6: Divide both sides: \(\frac{6x^{2}}{6}=\frac{216}{6}\) - step7: Divide the numbers: \(x^{2}=36\) - step8: Simplify the expression: \(x=\pm \sqrt{36}\) - step9: Simplify: \(x=\pm 6\) - step10: Separate into possible cases: \(\begin{align}&x=6\\&x=-6\end{align}\) - step11: Rewrite: \(x_{1}=-6,x_{2}=6\) The critical points are \( x = -6 \) and \( x = 6 \). Now, we need to determine which of these points correspond to the absolute maximum. We will evaluate the function at each critical point and compare the values. Evaluate the expression by following steps: - step0: Evaluate: \(\frac{6x}{x^{2}+36}\) - step1: Substitute: \(\frac{6\left(-6\right)}{\left(-6\right)^{2}+36}\) - step2: Multiply the numbers: \(\frac{-36}{\left(-6\right)^{2}+36}\) - step3: Add the numbers: \(\frac{-36}{72}\) - step4: Reduce the fraction: \(\frac{-1}{2}\) - step5: Rewrite the fraction: \(-\frac{1}{2}\) Substitute \( x=6 \) into the expression \( \frac{6x}{x^{2}+36} \). Evaluate the expression by following steps: - step0: Evaluate: \(\frac{6x}{x^{2}+36}\) - step1: Substitute: \(\frac{6\times 6}{6^{2}+36}\) - step2: Multiply the numbers: \(\frac{36}{6^{2}+36}\) - step3: Add the numbers: \(\frac{36}{72}\) - step4: Reduce the fraction: \(\frac{1}{2}\) The function values at the critical points are: - At \( x = -6 \), the function value is \( -\frac{1}{2} \). - At \( x = 6 \), the function value is \( \frac{1}{2} \). Since the function value at \( x = 6 \) is greater than the function value at \( x = -6 \), the absolute maximum point is at \( x = 6 \). Therefore, the coordinates of the absolute maximum point are \( (6, \frac{1}{2}) \).