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Differential Calculus

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2 Jan 2012 19:27:22 IST

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DIscuss the maxima and minima of the function f(x)=!x-2!+!x-5!+!x-3! (! is modulus)

Mathematics

DIscuss the maxima and minima of the function f(x)=!x-2!+!x-5!+!x-3! (! is modulus)

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vibhu mittal

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3 Jan 2012 16:04:37 IST

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The critical points of this function are 2,5,3 so check the value of f(x) at 2,3,5 and find maxima and minima....

jagdish singh

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16 Jan 2012 10:29:41 IST

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$\hspace{-16}$Let $\mathbf{y=f(x)=\mid x-2 \mid +\mid x-3 \mid+\mid x-5 \mid}$\\\\ $\mathbf{y=\underbrace{\bold{\mid x-2 \mid+\mid 5-x \mid}}+\mid x-3 \mid}$\\\\ Using $\mathbf{\triangle}$ Inequality in underbrace portion::\\\\ Which is $\mathbf{\mid x-a \mid+\mid b-x \mid\geq \mid b-a \mid}$\\\\ Inequality Hold When $\mathbf{(x-a).(b-x)\geq 0\Leftrightarrow a\leq x\leq b}$\\\\ So $\mathbf{y=\underbrace{\bold{\mid x-2 \mid+\mid 5-x \mid}}+\mid x-3 \mid\geq \mid 3\mid+\mid x-3 \mid\geq 0}$\\\\ So $\mathbf{y\geq 3\Leftrightarrow y\in [3,\infty)}$\\\\ Minimum occur at $\mathbf{x=3}$$

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