Find the location of the local extrema of the following function. \( f(x)=3-x^{\frac{2}{9}} \)

The function \( f(x) = 3 - x^{\frac{2}{9}} \) does not have any local extrema. To find the local extrema of a function, we need to find the critical points by taking the derivative of the function and setting it equal to zero. Let's find the derivative of the function \( f(x) = 3 - x^{\frac{2}{9}} \) with respect to \( x \) and then set it equal to zero to find the critical points. Evaluate the derivative by following steps: - step0: Evaluate the derivative: \(f^{\prime}\left(x\right)=\frac{d}{dx}\left(3-x^{\frac{2}{9}}\right)\) - step1: Use differentiation rules: \(f^{\prime}\left(x\right)=\frac{d}{dx}\left(3\right)-\frac{d}{dx}\left(x^{\frac{2}{9}}\right)\) - step2: Find the derivative: \(f^{\prime}\left(x\right)=0-\frac{2}{9}x^{-\frac{7}{9}}\) - step3: Remove 0: \(f^{\prime}\left(x\right)=-\frac{2}{9}x^{-\frac{7}{9}}\) - step4: Simplify: \(f^{\prime}\left(x\right)=-\frac{2\sqrt[9]{x^{2}}}{9x}\) The derivative of the function \( f(x) = 3 - x^{\frac{2}{9}} \) with respect to \( x \) is \( f'(x) = -\frac{2\sqrt[9]{x^{2}}}{9x} \). To find the critical points, we set the derivative equal to zero and solve for \( x \): \[ -\frac{2\sqrt[9]{x^{2}}}{9x} = 0 \] Solving this equation will give us the critical points of the function. Let's solve it. Solve the equation by following steps: - step0: Solve for \(x\): \(-\frac{2\sqrt[9]{x^{2}}}{9x}=0\) - step1: Find the domain: \(-\frac{2\sqrt[9]{x^{2}}}{9x}=0,x\neq 0\) - step2: Rewrite the expression: \(\frac{-2\sqrt[9]{x^{2}}}{9x}=0\) - step3: Cross multiply: \(-2\sqrt[9]{x^{2}}=9x\times 0\) - step4: Simplify the equation: \(-2\sqrt[9]{x^{2}}=0\) - step5: Change the sign: \(2\sqrt[9]{x^{2}}=0\) - step6: Rewrite the expression: \(\sqrt[9]{x^{2}}=0\) - step7: Set the radicand equal to 0: \(x^{2}=0\) - step8: Set the base equal to 0: \(x=0\) - step9: Check if the solution is in the defined range: \(x=0,x\neq 0\) - step10: Find the intersection: \(x \in \varnothing \) The solution to the equation \( -\frac{2\sqrt[9]{x^{2}}}{9x} = 0 \) is \( x \in \varnothing \), which means there are no real solutions for \( x \). Therefore, the function \( f(x) = 3 - x^{\frac{2}{9}} \) does not have any local extrema.