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 $(– \infty, 2)$
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