Show that f x = x 2 - 4 x is an increasing function.

Given that, we have: f x = x 2 - 4 x Now, differentiate f x with respect to x , then we get: f ' x = d d x x 2 - 4 x ⇒ f ' x = x d d x x 2 - 4 - x 2 - 4 d d x x x 2 [Using Quotient Rule in Differentiation] ⇒ f ' x = x 2 x - x 2 - 4 x 2 ⇒ f ' x = 2 x 2 - x 2 + 4 x 2 ⇒ f ' x = x 2 + 4 x 2 We know that, a function f x is increasing f ' x > 0 So, clearly, we have numerator is always positive and also, denominator is always positive. Therefore, the given function is an increasing function.

A function f is defined as f x = ( 2 x + 5 ) 2 - 3 for x ≥ 0 . Find an expression for f ' ( x ) and explain why f is an increasing function.

Given that, we have: f x = ( 2 x + 5 ) 2 - 3 , for x ≥ 0 Now, differentiate f x with respect to x , then we get: f ' x = d d x ( 2 x + 5 ) 2 - 3 ⇒ f ' x = d d x ( 2 x + 5 ) 2 - d d x 3 ⇒ f ' x = 2 ( 2 x + 5 ) d d x ( 2 x + 5 ) ⇒ f ' x = 2 × 2 2 x + 5 = 8 x + 20 We know that, a function f x is an increasing if f ' x > 0 Therefore, 8 x + 20 > 0 ⇒ 2 x + 5 > 0 ⇒ 2 x + 5 > 0 Critical value is - 5 2 and the function is increasing for x > - 5 2 Hence, the function is increasing for all the values of x in the domain of f x

A function f is defined as f x = 2 x 4 - x 2 for x > 0 . Show that f is a decreasing function.

Given that, we have: f x = 2 x 4 - x 2 , for x > 0 Now, differentiate f x with respect to x , then we get: f ' x = d d x 2 x 4 - x 2 ⇒ f ' x = 2 d d x 1 x 4 - d d x x 2 ⇒ f ' x = 2 - 4 x 5 - 2 x ⇒ f ' x = - 8 x 5 - 2 x ⇒ f ' x = - 8 - 2 x 6 x 5 ⇒ f ' x = - 8 + 2 x 6 x 5 We know that, a function f x is decreasing if f ' x < 0 Thus, for x > 0 , we have - 8 + 2 x 6 x 5 < 0 Hence, the function is decreasing for all the values of x in the domain of f x

E M B I B E

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8A>Q 2

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AS and A Level

IMPORTANT

Earn 100

Find the set of values of $x$ for which the following function is decreasing.

$f (x) = - 40 x + 13 x^{2} - x^{3}$ .

Important Questions on Further Differentiation

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8A>Q 2

Find the set of values of $x$ for which the following function is decreasing.

$f (x) = 11 + 24 x - 3 x^{2} - x^{3}$ .

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8A>Q 3

Find the set of values of $x$ for which the following function is increasing.

$f (x) = \frac{1}{6} (5 - 2 x)^{3} + 4 x$ .

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8A>Q 4

A function $f$ is defined as $f (x) = \frac{4}{1 - 2 x} for x \geq 1$ . Find an expression for $f^{'} (x)$ and determine whether $f$ is an increasing function, a decreasing function or neither.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8A>Q 5

A function $f$ is defined as $f (x) = \frac{5}{(x + 2)^{2}} - \frac{2}{x + 2} for x \geq 0$ . Find an expression for $f^{'} (x)$ and determine whether $f$ is an increasing function, a decreasing function or neither.

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AS and A Level

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Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8A>Q 6

Show that $f (x) = \frac{x^{2} - 4}{x}$ is an increasing function.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8A>Q 7

A function $f$ is defined as $f (x) = (2 x + 5)^{2} - 3$ for $x \geq 0$ . Find an expression for $f^{'} (x)$ and explain why $f$ is an increasing function.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8A>Q 8

A function $f$ is defined as $f (x) = \frac{2}{x^{4}} - x^{2}$ for $x > 0$ . Show that $f$ is a decreasing function.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8A>Q 9

A manufacturing company produces $x$ articles per day. The profit function, $P (x)$ , can be modelled by the function $P (x) = 2 x^{3} - 81 x^{2} + 840 x$ . Find the range of values of $x$ for which the profit is decreasing.