If x lies in the interval [0,1], then the least value of x2 + x + 1 is _______________ .

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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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