CAT 2025 Lesson : Functions - Minimum & Maximum

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

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: $20$

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:($2$) $y= a + bx + cx^2$