Find the intervals in which the function f x = 5 x 3 2 - 3 x 5 2 is increasing or decreasing.

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 = 5 x 3 2 - 3 x 5 2 f ' x = 15 2 x 1 2 - 15 2 x 3 2 = 15 2 x 1 2 1 - x Here, x = 0 , x = 1 are the critical points. The possible intervals are ( - ∞ , 0 ) , ( 0 , 1 ) and ( 1 , ∞ ) . . . . . ( 1 ) For f ( x ) to be increasing, we must have f ' x > 0 ⇒ 15 2 x 1 2 1 - x > 0 ⇒ x ∈ 0 , 1 From eq . ( 1 ) So, f x is increasing on x ∈ 0 , 1 . For f ( x ) to be decreasing, we must have f ' x < 0 ⇒ 15 2 x 1 2 1 - x < 0 ⇒ x ∈ 1 , ∞ From eq . ( 1 ) So, f ( x ) is decreasing on x ∈ 1 , ∞ .

Find the intervals in which the function f x = x 4 4 - x 3 - 5 x 2 + 24 x + 12 is increasing or decreasing.

Given function is f x = x 4 4 - x 3 - 5 x 2 + 24 x + 12 Differentiating with respect to x , we get: f ' x = 4 x 3 4 - 3 x 2 - 10 x + 24 ⇒ f ' ( x ) = x 3 - 3 x 2 - 10 x + 24 For, critical points put f ' ( x ) = 0 ∴ x 3 - 3 x 2 - 10 x + 24 = 0 ⇒ ( x - 2 ) x 2 - x - 12 = 0 ⇒ ( x - 2 ) x 2 - 4 x + 3 x - 12 = 0 ⇒ ( x - 2 ) x x - 4 + 3 x - 4 = 0 ⇒ ( x - 2 ) ( x - 4 ) ( x + 3 ) = 0 ⇒ x = 2 , 4 , - 3 Therefore, we have the intervals ( - ∞ , - 3 ) , ( - 3 , 2 ) ( 2 , 4 ) and ( 4 , ∞ ) Since, f ' ( x ) > 0 in ( - 3 , 2 ) ∪ ( 4 , ∞ ) Therefore, f x is strictly increasing in interval ( - 3 , 2 ) ∪ ( 4 , ∞ ) . And f ' ( x ) < 0 in ( - ∞ , - 3 ) ∪ ( 2 , 4 ) Therefore, f x is strictly decreasing in ( - ∞ , - 3 ) ∪ ( 2 , 4 ) .

E M B I B E

MATHEMATICS CLASS XII VOLUME-1>Chapter 17 - Increasing and Decreasing Functions>EXERCISE>Q 1

HARD

12th CBSE

IMPORTANT

Earn 100

Find the interval in which the following function are increasing or decreasing $f (x) = x^{3} - 12 x^{2} + 36 x + 17$ .

Important Questions on Increasing and Decreasing Functions

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 17 - Increasing and Decreasing Functions>EXERCISE>Q 1

Find the interval in which the following function are increasing or decreasing $f (x) = 2 x^{3} - 24 x + 7$ .

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 17 - Increasing and Decreasing Functions>EXERCISE>Q 1

Find the intervals in which the function

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

increasing or decreasing.

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 17 - Increasing and Decreasing Functions>EXERCISE>Q 1

Find the intervals in which the function

f (x) = 5 x^{\frac{3}{2}} - 3 x^{\frac{5}{2}}

is increasing or decreasing.

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 17 - Increasing and Decreasing Functions>EXERCISE>Q 1

Find the intervals in which the function

f (x) = x^{8} + 6 x^{2}

is increasing or decreasing.

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 17 - Increasing and Decreasing Functions>EXERCISE>Q 1

Find the intervals in which the function

f (x) = {\{x (x - 2)\}}^{2}

is increasing or decreasing.

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 17 - Increasing and Decreasing Functions>EXERCISE>Q 1

Find the intervals in which the function

f (x) = \frac{x^{4}}{4} - x^{3} - 5 x^{2} + 24 x + 12

is increasing or decreasing.

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 17 - Increasing and Decreasing Functions>EXERCISE>Q 1

Find the interval in which the following function are increasing or decreasing $f (x) = \frac{3}{10} x^{4} - \frac{4}{5} x^{3} - 3 x^{2} + \frac{36}{5} x + 11$ .