Question
Find the intervals in which the function f given by $f(x) = x^2 – 4x + 6$ is
- increasing
- decreasing
✓
Answer
We have
$f (x) = x^2 – 4x + 6$
or $f ′(x) = 2x – 4$
Therefore, $f ′(x) = 0$ gives $x = 2$.
Now the point $x = 2$ divides the real line into two disjoint intervals namely, $(– \infty, 2)$ and $(2, \infty).$
In the interval $(– \infty, 2), f ′(x) = 2x – 4 < 0.$
And in interval $(2, \infty ), f^\prime(x)=2x-4>0$
$\therefore (i) f $ is increasing in $(2, \infty$)
and $(ii)$ f is decreasing in $(–\infty, 2)$
$f (x) = x^2 – 4x + 6$
or $f ′(x) = 2x – 4$
Therefore, $f ′(x) = 0$ gives $x = 2$.
Now the point $x = 2$ divides the real line into two disjoint intervals namely, $(– \infty, 2)$ and $(2, \infty).$
In the interval $(– \infty, 2), f ′(x) = 2x – 4 < 0.$
And in interval $(2, \infty ), f^\prime(x)=2x-4>0$
$\therefore (i) f $ is increasing in $(2, \infty$)
and $(ii)$ f is decreasing in $(–\infty, 2)$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Find the principal value of $\tan^{-1}\ (-1).$
→The area bounded by the curve y = x|x|, x-axis and the ordinates x = -1 and x = 1 is given by
If $\overrightarrow{\text{a}} = 2\hat{\text{i}} + \hat{\text{j}} + 3\hat{\text{k}}$ and $ \overrightarrow{\text{b}} = 3\hat{\text{i}}+ 5\hat{\text{j}} - 2\hat{\text{k}},$ then find $|\overrightarrow{\text{a}}\times|\overrightarrow{\text{b}|}.$
→Find the distance between the points P(1, -3, 4) and Q (-4, 1, 2)
Write the number of vectors of unit length perpendicular to both the vectors $\overrightarrow{\text{a}} = 2\hat{\text{i}} + \hat{j} + 2\hat{\text{k}} \text{ and} \overrightarrow{\text{b}} = \hat{\text{j}} + \hat{\text{k}}.$
→Write the order of the differential equation of all non-horizontal lines in a plane.
Evaluate the following:
$\tan^{-1}(\tan4)$
→$\tan^{-1}(\tan4)$
Solve the following matrix equation for $\text{x};[\text{x } { 1 }] \ \begin{bmatrix}1 & 0\\-2 & 0\\\end{bmatrix} \ = \text{O}.$
→Using principal value, evaluate the following:$\sin^{-1} \bigg( \sin \frac{3{\pi}}{5}\bigg)$
→If $\text{A}[\text{a}_{\text{ij}}]=\begin{bmatrix}2&3&-5\\1&4&9\\0&7&-2\end{bmatrix}$ and $\text{B}=[\text{b}_\text{ij}]=\begin{bmatrix}2&-1\\-3&4\\1&-2\end{bmatrix}$
Then find $a_{22} + b_{21}$
→Then find $a_{22} + b_{21}$