\( y=\frac{6 x}{x^{2}+36} \) Find the inflection points. Select the cor

Analyze the function features with analysis_type = "inflection_points" and expression = "y=6x/(x^2+36)". Function by following steps: - step0: Find the inflection points: \(y=\frac{6x}{x^{2}+36}\) - step1: Use the appropriate notation: \(f\left(x\right)=\frac{6x}{x^{2}+36}\) - step2: Find the domain: \(f\left(x\right)=\frac{6x}{x^{2}+36},x \in \mathbb{R}\) - step3: Find the derivative: \(f^{\prime}\left(x\right)=\frac{-6x^{2}+216}{\left(x^{2}+36\right)^{2}}\) - step4: Find the second derivative: \(f^{\prime\prime}\left(x\right)=\frac{12x^{3}-1296x}{\left(x^{2}+36\right)^{3}}\) - step5: Find the domain: \(f^{\prime\prime}\left(x\right)=\frac{12x^{3}-1296x}{\left(x^{2}+36\right)^{3}},x \in \mathbb{R}\) - step6: Substitute \(f^{\prime\prime}\left(x\right)=0:\) \(0=\frac{12x^{3}-1296x}{\left(x^{2}+36\right)^{3}}\) - step7: Swap the sides: \(\frac{12x^{3}-1296x}{\left(x^{2}+36\right)^{3}}=0\) - step8: Cross multiply: \(12x^{3}-1296x=\left(x^{2}+36\right)^{3}\times 0\) - step9: Simplify the equation: \(12x^{3}-1296x=0\) - step10: Factor the expression: \(12x\left(x^{2}-108\right)=0\) - step11: Divide both sides: \(x\left(x^{2}-108\right)=0\) - step12: Separate into possible cases: \(\begin{align}&x=0\\&x^{2}-108=0\end{align}\) - step13: Solve the equation: \(\begin{align}&x=0\\&x=6\sqrt{3}\\&x=-6\sqrt{3}\end{align}\) - step14: Determine the intervals: \(\begin{align}&\left(-6\sqrt{3},0\right),\left(0,6\sqrt{3}\right)\\&\left(0,6\sqrt{3}\right),\left(6\sqrt{3},+\infty\right)\\&\left(-\infty,-6\sqrt{3}\right),\left(-6\sqrt{3},0\right)\end{align}\) - step15: Choose the points: \(\begin{align}&x_{1}=-5,x_{2}=5\\&x_{3}=5,x_{4}=11\\&x_{5}=-11,x_{6}=-5\end{align}\) - step16: Find the value of the second derivative: \(\begin{align}&f^{\prime\prime}\left(-5\right)\approx 0.02194,f^{\prime\prime}\left(5\right)\approx -0.02194\\&f^{\prime\prime}\left(5\right)\approx -0.02194,f^{\prime\prime}\left(11\right)\approx 0.000443\\&f^{\prime\prime}\left(-11\right)\approx -0.000443,f^{\prime\prime}\left(-5\right)\approx 0.02194\end{align}\) - step17: The inflection point is at \(x=0:\) \(\begin{align}&f\left(x\right)=\frac{6x}{x^{2}+36},x=0\\&f\left(x\right)=\frac{6x}{x^{2}+36},x=6\sqrt{3}\\&f\left(x\right)=\frac{6x}{x^{2}+36},x=-6\sqrt{3}\end{align}\) - step18: Calculate the function value: \(\begin{align}&f\left(0\right)=0\\&f\left(6\sqrt{3} \right)=\frac{\sqrt{3}}{4}\\&f\left(-6\sqrt{3} \right)=-\frac{\sqrt{3}}{4}\end{align}\) - step19: The inflection point is \(\left(0,0\right):\) \(\begin{align}&\left(0,0\right)\\&\left(6\sqrt{3},\frac{\sqrt{3}}{4}\right)\\&\left(-6\sqrt{3},-\frac{\sqrt{3}}{4}\right)\end{align}\) The inflection points of the function \( y=\frac{6x}{x^{2}+36} \) are at the points \((0,0)\), \((6\sqrt{3},\frac{\sqrt{3}}{4})\), and \((-6\sqrt{3},-\frac{\sqrt{3}}{4})\).