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Sign of the Derivative Theory
![For a given function \(y=f(x)\) if \(\dfrac{d y}{d x}>0\) in a given domain \(a<x<b\) then the function is increasing in that domain.\\ Conversely if \(\dfrac{d y}{d x}<0\) then the function is decreasing in that domain.\\ \textbf{Example}\\ Find the values of \(x\) for which the curve \(y=x^3+12 x^2+45 x-30\) is \begin{itemize} \item[\bf{i)}]increasing \item[\bf{ii)}]decreasing \end{itemize} \textbf{Solution}\\ $\begin{aligned} \text { (i) }\quad y&=x^3+12 x^2+45 x-30 \\ \frac{d y}{d x}&=3 x^2-24 x+45 \end{aligned}$\\ $ \begin{aligned} \text { For }\frac{d y}{d x}>0 \quad 3 x^2-24 x+45&>0 \\ x^2-8 x+15&>0 \\ (x-3)(x-5)&>0 \\ \therefore x<3 \cup x>5 \quad &\text { the curve increases. } \end{aligned}$\\ (ii) For \(3<x<5\) the curve decreases\\](/media/tcdgsbss/4861.png)
NSW Y12 Maths - Advanced Differential Calculus Sign of the Derivative
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