Question

The absolute minimum value, of the function $f(x)=\left|x^2-x+1\right|+\left[x^2-x+1\right]$, where $[t]$ denotes the greatest integer function, in the interval $[-1,2]$, is:

• A. $\frac{3}{2}$
• B. $\frac{1}{4}$
• C. $\frac{5}{4}$
• D. $\frac{3}{4}$

Answer 1

Answer: (D)

$\frac{3}{4}$

Answer 2

f(x)=∣∣​x2−x+1∣∣​+[x2−x+1];x∈[−1,2]
Let g(x)=x2−x+1
=(x−21​)2+43​
∵∣∣​x2−x+1∣∣​ and [x2−x+2]
Both have minimum value at x=1/2
⇒ Minimum f(x)=43​+0
=43​
So, the correct option is (D) : $\frac{3}{4}$