Find the local maxima and local minima of the function $\displaystyle f(x)=\frac{8}{3} x^{3}-12 x^{2}+18 x+5$.

\[\begin{aligned} & f(x)=\frac{8}{3} x^{3}-12 x^{2}+18 x+5 \\ & \Rightarrow f^{\prime}(x)=8 x^{2}-24 x+18 \\ & \quad=2\left(4 x^{2}-12 x+9\right)=2(2 x-3)^{2} \end{aligned} \] For critical points, Put $\displaystyle f(x)=0$ \[\Rightarrow 2(2 x-3)^{2}=0 \Rightarrow x=\frac{3}{2} \] since $\displaystyle f^{\prime}(x)$ does not change the sign as crosses $\displaystyle x=\frac{3}{2}$ from left to right, $\displaystyle f$ has no local maxima or local minima.