Find the intervals on which the following function f ( x ) = 3 10 x 4 - 4 5 x 3 - 3 x 2 + 36 5 x + 11 is strictly increasing.

Given that f ( x ) = 3 10 x 4 - 4 5 x 3 - 3 x 2 + 36 5 x + 11 Differentiating f ( x ) , we get f ' ( x ) = 1 10 ( 12 x 3 - 24 x 2 - 60 x + 72 ) ⇒ f ' ( x ) = 12 10 x 3 - 2 x 2 - 5 x + 6 ⇒ f ' ( x ) = 12 10 ( x - 1 ) ( x - 3 ) ( x + 2 ) Now, f ' ( x ) = 0 ⇒ x = 1 or x = 3 or x = - 2 . These points divides the whole real line in four disjoint open intervals, namely, ( - ∞ , - 2 ) , ( - 2 , 1 ) , ( 1 , 3 ) and ( 3 , ∞ ) . Case i) When x ∈ ( - ∞ , - 2 ) In this case f ' ( x ) = ( - ) ( - ) ( - ) = - ve Case ii) When x ∈ ( - 2 , 1 ) In this case f ' ( x ) = ( - ) ( - ) ( + ) = + ve Case iii) When x ∈ ( 1 , 3 ) In this case f ' ( x ) = ( + ) ( - ) ( + ) = - ve Case iv) when x ∈ ( 3 , ∞ ) In this case f ' ( x ) = ( + ) ( + ) ( + ) = + ve. ∴ f ' ( x ) > 0 on - 2 , 1 and ( 3 , ∞ ) . Hence f ( x ) is strictly increasing on intervals ( - 2 , 1 ) and 3 , ∞ .

Find the intervals on which the following function f ( x ) = 3 10 x 4 - 4 5 x 3 - 3 x 2 + 36 5 x + 11 is strictly decreasing.

Given that f ( x ) = 3 10 x 4 - 4 5 x 3 - 3 x 2 + 36 5 x + 11 Differentiating f ( x ) , we get f ' ( x ) = 1 10 ( 12 x 3 - 24 x 2 - 60 x + 72 ) ⇒ f ' ( x ) = 12 10 x 3 - 2 x 2 - 5 x + 6 ⇒ f ' ( x ) = 12 10 ( x - 1 ) ( x - 3 ) ( x + 2 ) Now, f ' ( x ) = 0 ⇒ x = 1 or x = 3 or x = - 2 . These points divides the whole real line in four disjoint open intervals, namely, ( - ∞ , - 2 ) , ( - 2 , 1 ) , ( 1 , 3 ) and ( 3 , ∞ ) . Case i) When x ∈ ( - ∞ , - 2 ) In this case f ' ( x ) = ( - ) ( - ) ( - ) = - ve Case ii) When x ∈ ( - 2 , 1 ) In this case f ' ( x ) = ( - ) ( - ) ( + ) = + ve Case iii) When x ∈ ( 1 , 3 ) In this case f ' ( x ) = ( + ) ( - ) ( + ) = - ve Case iv) when x ∈ ( 3 , ∞ ) In this case f ' ( x ) = ( + ) ( + ) ( + ) = + ve. ∴ f ' ( x ) < 0 on - ∞ , - 2 and ( 1 , 3 ) . Hence f ( x ) is strictly decreasing on intervals - ∞ , - 2 and ( 1 , 3 ) .

E M B I B E

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 11 - Applications of Derivatives>EXERCISE>Q 30

MEDIUM

12th CBSE

IMPORTANT

Earn 100

Find the intervals on which the following function $f (x) = x^{4} - 8 x^{3} + 22 x^{2} - 24 x + 21$ is decreasing.

Important Questions on Applications of Derivatives

MEDIUM

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 11 - Applications of Derivatives>EXERCISE>Q 31

Find the intervals on which the following function $f (x) = 3 x^{4} - 4 x^{3} - 12 x^{2} + 5$ is strictly increasing.

MEDIUM

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 11 - Applications of Derivatives>EXERCISE>Q 31

Find the intervals on which the following function $f (x) = 3 x^{4} - 4 x^{3} - 12 x^{2} + 5$ is strictly decreasing.

MEDIUM

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 11 - Applications of Derivatives>EXERCISE>Q 32

Find the intervals on which the following function $f (x) = \frac{3}{10} x^{4} - \frac{4}{5} x^{3} - 3 x^{2} + \frac{36}{5} x + 11$ is strictly increasing.

MEDIUM

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 11 - Applications of Derivatives>EXERCISE>Q 32

Find the intervals on which the following function $f (x) = \frac{3}{10} x^{4} - \frac{4}{5} x^{3} - 3 x^{2} + \frac{36}{5} x + 11$ is strictly decreasing.