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NSW Y12 Maths - Advanced Differential Calculus Global Maxima and Minima

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Global Maxima and Minima Theory

For \(y=f(x)\) in the domain \(a \leqslant x \leqslant b\), \textbf{the global maximum} is the greatest value in the domain which may be the local maximum on \(y=f(a)\) or \(y=f(b)\). \\ \textbf{The global minimum} is the lowest value in the domain which may include the local minimum on \(y=f(a)\) or \(y=f(b)\)\\ \textbf{Example}\\ %17215 The global maximum of the function \(y = - {x^3} - 3{x^2} + 9x\) in the domain \( - 4 \le x \le 2\) is ?\\ \textbf{Solution}\\ $\begin{aligned} &\begin{array}{l} y=-x^{3}-3 x^{2}+9 x \\ y^{\prime}=-3 x^{2}-6 x+9 \end{array}\\ &\text{Let}\ y^{\prime}=0\ \text{to locate stationary points}\\ &-3 x^{2}-6 x+9=0\\ &-3\left(x^{2}+2 x-3\right)=0\\ &-3(x-1)(x+3)=0\\ &x=1 \ \text{or}\ x=-3\\ &x=1,\ y=-1-3+9=5\\ &x=-3,\ y=27-27-27=-27\\ &\begin{aligned} x=-4,\,y &=64-48-36 \\ &=-20 \\ x=2, y &=-8-12+18 \\ &=-2 \end{aligned}\\ &\therefore\ \text{Global maximum} =5 \end{aligned}$\\

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  • Global Maxima and Minima - Video - Introduction to global maxima and minima

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