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Question
If x lies in the interval [0,1], then the least value of x2 + x + 1 is _______________ .
Options
3
`3/4`
1
none of these
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Solution
1
\[\text { Given: } f\left( x \right) = x^2 + x + 1\]
\[ \Rightarrow f'\left( x \right) = 2x + 1\]
\[\text { For a local maxima or a local minima, we must have } \]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 2x + 1 = 0\]
\[ \Rightarrow 2x = - 1\]
\[ \Rightarrow x = \frac{- 1}{2} \not\in \left[ 0, 1 \right]\]
\[\text { At extreme points } : \]
\[ f\left( 0 \right) = 0\]
\[f\left( 1 \right) = 1 + 1 + 1 = 3 > 0\]
\[\text { So, x = 1 is a local minima }. \]
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