Determine the intervals in which the function f (x) = x^{4} – 8x^{3} + 22x^{2} – 24x + 21 is decreasing or increasing

### Asked by Topperlearning User | 6th Aug, 2014, 10:17: AM

We have,

*f* (*x*) = *x*^{4} – 8*x*^{3} + 22*x*^{2} – 24*x* + 21.

*f *¢ (*x*) = 4*x*^{3} – 24*x*^{2} + 44*x* – 24 = 4(*x*^{3} – 6*x*^{2} + 11 *x* – 6)

*f *¢ (*x*) = 4(*x* – 1) (*x*^{2} – 5*x* + 6)

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

*f *¢ (*x*) > 0

Þ 4(*x *– 1) (*x*^{2} – 5*x* + 6) > 0

Þ (*x* – 1) (*x*^{2} – 5*x* + 6) > 0

Þ (*x* – 1) (*x* – 2)(*x* – 3) > 0 [_{} 4 > 0]

Þ 1 < *x* < 2 or 3 < *x* < ¥

Þ *x* Î (1, 2) È (3, ¥)

So, *f* (*x*) is increasing on (1, 2) È (3, ¥)

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

*f *¢ (*x*) < 0

Þ 4(*x* – 1) (*x*^{2} – 5*x* + 6) > 0

Þ (*x* – 1)(*x*^{2} – 5*x* + 6) < 0 [_{} 4 > 0]

Þ (*x* – 1) (*x* – 2) (*x* – 3) < 0

Þ 2 < *x* < 3 or *x* < 1

Þ *x* Î (2, 3) È (–¥, 1)

So, *f* (*x*) is decreasing on (2, 3) È (–¥, 1)

### Answered by | 6th Aug, 2014, 12:17: PM

## Concept Videos

- A man is moving away from a 40 m high tower at a speed of 2m/s . find the rate at which angle of elevation of the top of the tower is changing when he is at a distance of 30m from foot of tower. Assume that eye level of the man is 1.6 m from the ground. explain in great detail
- Show that the function f (x) = 2x + 3 is strictly increasing function on R.
- Show that the function f (x) = x
^{2}is strictly increasing function on [0,∞). - Show that the function f (x) = a
^{x}, a > 1 is strictly increasing on R. - Find the intervals in which f (x) = –x
^{2}– 2x + 15 is increasing or decreasing - Find the intervals in which the function f (x) is increasing, f (x) = 2x
^{3}- 9x^{2}+ 12x + 15 - Find the intervals in which the function f (x) is decreasing f(x) = 2x
^{3}+ 9x^{2}+ 12x + 20 - Find the intervals in which f (x) = (x + 1)
^{3}(x - 3)^{3}is increasing or decreasing. - Find the intervals in which the function f (x) =
_{}is increasing or decreasing

### Kindly Sign up for a personalised experience

- Ask Study Doubts
- Sample Papers
- Past Year Papers
- Textbook Solutions

#### Sign Up

#### Verify mobile number

Enter the OTP sent to your number

Change