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Which one of the following statements is correct?
Consider the function $\mathrm{f}(\mathrm{x})=|\mathrm{x}-1|+\mathrm{x}^{2}$, where $\mathrm{x} \in \mathbf{R}$.
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Consider the function $\mathrm{f}(\mathrm{x})=|\mathrm{x}-1|+\mathrm{x}^{2}$, where $\mathrm{x} \in \mathbf{R}$.
Solution:
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Verified Answer
The correct answer is:
$\mathrm{f}(\mathrm{x})$ has local minimum at one point only in $(-\infty, \infty)$
$f(x)$ has local minimum at one point only in $(-\infty, \infty$ $f^{\prime}(x)=\left\{\begin{array}{ll}2 x-1 & ; \quad x < 1 \\ 2 x+1 & ; \quad x \geq 1\end{array}\right.$
Clearly; for $(\mathrm{x}>1) ; \mathrm{f}^{\prime}(\mathrm{x})>0 \geq \&$ for $(\mathrm{x} < 1)$
$\mathrm{x}=\frac{1}{2}$ is the point of local minima
Clearly; for $(\mathrm{x}>1) ; \mathrm{f}^{\prime}(\mathrm{x})>0 \geq \&$ for $(\mathrm{x} < 1)$
$\mathrm{x}=\frac{1}{2}$ is the point of local minima
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