Question

Find the intervals in which the function

f (x) = x^{4} - 4 x^{3} + 4 x^{2} + 15

increasing or decreasing.

Accepted Answer

When $(x - a) (x - b) > 0$ with $a < b, x < a$ or $x > b$

When $(x - a) (x - b) < 0$ with $a < b, a < x < b$

$f (x) = x^{4} - 4 x^{3} + 4 x^{2} + 15$

$f' (x) = 4 x^{3} - 12 x^{2} + 8 x$

$= 4 x (x^{2} - 3 x + 2)$

$= 4 x (x - 1) (x - 2)$

Here, $x = 0, x = 1$ and $x = 2$ are the critical points

The possible intervals are $(- \infty, 0), (0, 1), (1, 2) and (2, \infty) . . . . . (1)$

For $f (x)$ to be increasing, we must have

$f' (x) > 0$

$\Rightarrow 4 x (x - 1) (x - 2) > 0 [Since, 4 > 0, 4 x (x - 1) (x - 2) > 0 \Rightarrow x (x - 1) (x - 2) > 0]$

$\Rightarrow x (x - 1) (x - 2) > 0$

$\Rightarrow x \in (0, 1) \cup (2, \infty) [From eq . (1)]$

So, $f (x)$ is increasing on $x \in (0, 1) \cup (2, \infty)$ .

For $f (x)$ to be decreasing, we must have

$f' (x) < 0$

$\Rightarrow 4 x (x - 1) (x - 2) < 0 [Since 4 > 0, 4 x (x - 1) (x - 2) < 0 \Rightarrow x (x - 1) (x - 2) < 0]$

$\Rightarrow x (x - 1) (x - 2) < 0$

$\Rightarrow x \in (- \infty, 0) \cup (1, 2) [From eq . (1)]$

So, $f (x)$ is decreasing on $x \in (- \infty, 0) \cup (1, 2)$ .

R D Sharma Solutions for Chapter: Increasing and Decreasing Functions, Exercise 1: EXERCISE

R D Sharma Mathematics Solutions for Exercise - R D Sharma Solutions for Chapter: Increasing and Decreasing Functions, Exercise 1: EXERCISE

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