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NSW Y12 Maths - Advanced Differential Calculus Stationary and Turning Points

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Stationary and Turning Points Theory

For the function \(y=f(x)\) stationary points occur when \(f^{\prime}(x)=0\)\\ There and three types of stationary points\\ - If \(f^{\prime}(a)=0\) and \(f^{\prime}(a-\epsilon)>0\) where \(\epsilon\) is a small positive number and \(f^{\prime}(a+\epsilon)<0\) then a maximum stationary point occurs.\\ - If \(f^{\prime}(a)=0\) and \(f^{\prime}(a-\epsilon)<0\) and \(f^{\prime}(a+\epsilon)>0\) then a minimum stationary point occur.\\ - If \(f^{\prime}(a)=0\) and \(f^{\prime}(a-\epsilon)<0\) and \(f^{\prime}(a+\epsilon)<0\) OR \(f^{\prime}(a-\epsilon)>0\) and \(f^{\prime}(a+\epsilon)>0\) then a horizontal point of inflexion occurs.\\ \textbf{Example}\\ Find the stationary points on the curve \(f(x)=x^3-3 x^2+5\) and determine their nature\\ \textbf{Solution}\\ $\begin{aligned} f(x)&=x^3-3 x^2+5 \\ f^{\prime}(x)&=3 x^2-6 x \\ \end{aligned}$\\ $\begin{aligned} \text { Let } f^{\prime}(x)=0 \quad 3 x^2-6 x&=0 \\ 3x(x-2)&=0 \end{aligned}$\\ $\begin{aligned} &\therefore \text { Stationary points at }(0,5) \text { and }(2,1) \\ &\text { At }(0,5) \quad f^{\prime}(-0.1)>0 \text { and } f^{\prime}(0.1)<0 \\ &\therefore \text { maximum stationary point } \\ &\text { At }(2,1) \quad f^{\prime}(1.9)<0 \text { and } f^{\prime}(2.1)>0 \\ &\therefore \text { minimum stationary point. } \end{aligned}$\\

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  • Stationary and Turning Points - Video - Differentiation : How to Find Stationary Points

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