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Chapter 4: Problem 3
\(f(x)=x^{2} \ln g(x)\) \(f^{\prime}(x)=2 x \ln g(x)+\frac{x^{2} g^{\prime}(x)}{g(x)}\) \(f^{\prime}(2)=4 \ln 3-\frac{16}{3}\)
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\(x=a \cos t+\frac{b}{2} \cos 2 t, \quad y=a \sin t+\frac{b}{2} \sin 2 t\) $\frac{d x}{d t}=-a \sin t-b \sin 2 t, \quad \frac{d y}{d t}=a \cos t+b \cos 2 t$ \(\frac{d y}{d x}=\frac{-(a \cos t+b \cos 2 t)}{a \sin t+b \sin 2 t}\) $\frac{d^{2} y}{d x^{2}}=\left[\begin{array}{l}\frac{(a \sin t+b \sin 2 t)(+a \sin t+2 b \sin 2 t)}{(a \sin t+b \sin 2 t)^{2}} \\ +(a \cos t+b \cos 2 t)(a \cos t+2 b \cos 2 t)\end{array}\right] \times \frac{1}{-(a \sin t+b \sin 2 t)}$ For \(\frac{d^{2} y}{d x^{2}}=0\) \((a \sin t+b \sin 2 t)(a \sin t+2 b \sin 2 t)+(a \cos t+b \cos 2 t)\) \((a \cos t+2 b \cos 2 t)=0\) \(\Rightarrow a^{2}+2 b^{2}+a b \cos t+2 a b \cos t=0\) \(\Rightarrow \cos t=\frac{-\left(a^{2}+2 b^{2}\right)}{3 a b}\)
\((f(x))^{n}=f(n x)\) Differentiating it \((f(x))^{n-1} f^{\prime}(x)=f^{\prime}(n x)\) Multiply by \(f(x)\) $\mathrm{f}(\mathrm{nx}) \mathrm{f}^{\prime}(\mathrm{x})=\mathrm{f}^{\prime}(\mathrm{nx}) \mathrm{f}(\mathrm{x})$
$\begin{aligned} &a x^{3}+b x^{2}+b x+d=0\\\ &3 a x^{2}+2 b x+b=0\\\ &\Rightarrow 3 \mathrm{a}+3 \mathrm{~b}=0 \quad \text { (as } 1 \text { is repeated root) }\\\ &\Rightarrow a+b=0\\\ &\text { Now, a }+b+b+d=0\\\ &\Rightarrow \mathrm{b}+\mathrm{d}=0 \end{aligned}$
As $\mathrm{f}(\mathrm{x}), \mathrm{f}^{\prime}(\mathrm{x}), \mathrm{f}^{\prime \prime}(\mathrm{x})\( are all \)+\mathrm{ve} \forall \mathrm{x} \in[0,7]$ \(\Rightarrow \mathrm{f}(\mathrm{x})\) is increasing function and concave up. and so is \(\mathrm{f}^{-1}(\mathrm{x})\). $\Rightarrow f^{-1}(5)+4 f^{-1}\left(\frac{2 g}{5}\right)\( is always \)+v e$.
$\begin{aligned} &x^{2}+y^{2}=a^{2} \\ &2 x+2 y y^{\prime}=0 \Rightarrow \frac{x}{y}=-y^{\prime} \\ &1+\left(y^{\prime}\right)^{2}+y y^{\prime \prime}=0 \Rightarrow y=\frac{-\left(1+\left(y^{\prime}\right)^{2}\right)}{y^{\prime \prime}} \end{aligned}$ Using, (I), (II) \& (III) $\begin{aligned} &\left(\mathrm{y}^{\prime}\right)^{2}+1=\frac{\mathrm{a}^{2}\left(\mathrm{y}^{\prime \prime}\right)^{2}}{\left(1+\left(\mathrm{y}^{\prime}\right)^{2}\right)^{4}} \Rightarrow \frac{1}{\mathrm{a}^{2}}=\frac{\left(\mathrm{y}^{\prime \prime}\right)^{2}}{\left(1+\left(\mathrm{y}^{\prime}\right)^{2}\right)^{2}\left(1+\left(\mathrm{y}^{\prime}\right)^{2}\right)} \\\ &\Rightarrow \mathrm{K}=\frac{\left|\mathrm{y}^{\prime \prime}\right|}{\sqrt{\left(1+\left(\mathrm{y}^{\prime}\right)^{2}\right)^{3}}} \end{aligned}$
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