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

Answer 1

1

 

\[\text { Given: } f\left( x \right) = x^2 + x + 1\]

\[ \Rightarrow f'\left( x \right) = 2x + 1\]

\[\text { For a local maxima or a local minima, we must have } \]

\[f'\left( x \right) = 0\]

\[ \Rightarrow 2x + 1 = 0\]

\[ \Rightarrow 2x = - 1\]

\[ \Rightarrow x = \frac{- 1}{2} \not\in \left[ 0, 1 \right]\]

\[\text { At extreme points } : \]

\[ f\left( 0 \right) = 0\]

\[f\left( 1 \right) = 1 + 1 + 1 = 3 > 0\]

\[\text { So, x = 1 is a local minima }. \]