Prove that the function f given by f(x) = x^2 - x + 1 is neither … - Vidyadip

Question

Prove that the function f given by $f(x) = x^2 - x + 1$ is neither strictly increasing nor strictly decreasing on $(-1, 1)$.

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Answer

$\text{f}\text{(x)} = \text{x}^{2} - \text{x} + 1\ \Rightarrow\ \text{f}'\text{(x)} = 2\text{x} - 1$
f(x) is strictly increasing if f'(x) > 0 $\Rightarrow\ 2\text{x} - 1 > 0 \ \Rightarrow\ \text{x}>\frac{1}{2}$
i.e., increasing on the interval $\Big(\frac{1}{2},\ 1\Big)$
f(x) is strictly decreasing if f'(x) < 0 $ \Rightarrow\ 2\text{x} - 1< 0 \ \Rightarrow \ \text{x} < \frac{1} {2}$
i.e., decreasing on the interval $\Big(-1,\ \frac{1}{2}\Big)$
hence, f(x) is neither strictly increasing nor decreasing on the interval (-1, 1 ).

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