Sets, Relations & Functions
Absolute Value + Greatest Integer Function
nta_pyq_2023_jan
Grade 11
Question:
The absolute minimum value of the function $f(x) = |x^2 - x + 1| + [x^2 - x + 1]$, where $[t]$ denotes the greatest integer function, in the interval $[-1, 2]$, is:
$\dfrac{3}{4}$
$\dfrac{3}{2}$
$\dfrac{1}{4}$
$\dfrac{5}{4}$
Step-by-Step Solution
Key Concept: $|t|+[t] = 0$ when $-1 < t < 0$, and $= 2t$ when $t \ge 0$; minimum of $x^2-x+1$ is $3/4$ on $[-1,2]$.
Min of $t=x^2-x+1=3/4$ at $x=1/2$. Since $0<3/4<1$, $[t]=0$ and $f=|3/4|+0=3/4$.
Correct Answer: 1