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Algebra

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CAT 2025 Lesson : Functions - Minimum & Maximum

5.3 Minimum and Maximum Functions

These functions are expressed either as the minimum of

2

or more expressions, or as the maximum of

2

or more expressions.

We are typically given

2

linear expressions with one having a positive slope and the other having a negative slope. In these cases, the value of

x

at which a minimum function attains maximum or a maximum function attains minimum is when the expressions are equal.

Example 9

Let

f(x) = \text{min}(3x + 7, 15 – x)

. What is the value of

x

at which

f(x)

attains maximum?

Solution

3x + 7

is positively sloped, while

15 - x

is negatively sloped. Therefore, we will be able to find the maximum for this minimum function.

f(x)

will attain maximum when

3x + 7 = 15 - x

⇒

4x = 8

⇒

x = 2

Answer: 2

Example 10

Let

f(x) = \text{max} \left[5x, 52 - 2x^2 \right]

, where

x

is any positive real number. Then the minimum possible value of

f(x)

is
[CAT 2018 Shift 2]

Solution

To find the minimum of the maximum function, we equate the two terms to find the

x

-value.

5x = 52 - 2x^2

⇒

2x^2 + 5x - 52 = 0

⇒

2x^2 + 13x - 8x - 52 = 0

⇒

(x - 4)(2x + 13) = 0

⇒

x = 4

(The negative root is rejected as x is a positive real number)

We need to find the minimum value of

f(x)

f(4) = 5 \times 4 = 20

Answer:

20

5.4 Identifying Order of the Expression

Where

x

is the input and

f(x)

is the output, for consecutive inputs if

If	Then Function is a
Differences (D) are constant	Linear Expression (Highest Power = 1)
Difference of Difference (DoD) is constant	Quadratic Expression (Highest Power = 2)
DoDoD is constant	Cubic Expression (Highest Power = 3)
DoDoDoD is constant	Biquadratic Expression (Highest Power = 4)

and so on...

Example 11

In the below table, for suitably chosen constants

a

b

and

c

, which one of the following best describes the relation between

y

and

x

$x$	$1$	$2$	$3$	$4$	$5$	$6$
$y$	$4$	$8$	$14$	$22$	$32$	$44$

[CAT 2000]

(1)

y = a + bx

(2)

y= a + bx + cx^2

(3)

y=e^{a + bx}

(4) None of the above

Solution

As the

x

-values are consecutive, we shall start by checking the differences of consecutive

y

-values.

Differences

= (8 - 4), (14 - 8), (22 - 14), (32 - 22), (44 - 32) = 4, 6, 8, 10, 12

As these are not constant, we next check for DoD
Difference of Difference (DoD)

= (6 - 4), (8 - 6), (10 - 8), (12 - 10) = 2, 2, 2, 2

As all DoDs are equal, this is a quadratic function. Only Option

(2)

has a quadratic expression.

Answer:(

2

)

y= a + bx + cx^2

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