Application of Derivatives — Maths STD 12 Science — Question
Gujarat BoardEnglish MediumSTD 12 ScienceMathsApplication of Derivatives2 Marks
Question
Prove that the function f given by $f(x) = x^2 - x + 1$ is neither strictly increasing nor decreasing on $(-1, 1)$.
✓
Answer
It is given that function $f(x) = x^2 - x + 1$
$f'(x) = 2x - 1$
If $f'(x) = 0$, then we get
$~~~ ~~~x=\frac{1}{2}$
So, the point x = $\frac{1}{2}$ divides the interval (-1, 1) into two disjoint intervals, $\left(-1, \frac{1}{2}\right)$ and $\left(\frac{1}{2}, 1\right)$
In interval $\left(-1, \frac{1}{2}\right)$, we have
$f'(x) = 2x - 1 < 0$
Therefore, the given function 'f 'is strictly decreasing in interval $\left(-1, \frac{1}{2}\right)$
Also, in interval $\left(\frac{1}{2}, 1\right)$
$f'(x) = 2x - 1 > 0$
Hence, the given function 'f ' is strictly increasing in interval for $\left(\frac{1}{2}, 1\right)$.
Therefore, f is neither strictly increasing nor decreasing in interval (-1, 1).
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.