First slide

Functions (XII)

Question

Let $f (x) = | x - 1 | + | x - 2 | + | x - 3 | + | x - 4 |$ then

Moderate

A

least value of f(x) is 4
B

least value is not attained at unique point
C

the number of integral solution of f(x) = 4 is 2
D

the value of $\frac{f (π - 1) + f (e)}{2 f (\frac{12}{5})}$ is 1

Solution

$f (x) = \{\begin{array}{cl} 10 - 4 x, & if - \infty < x < 1 \\ 8 - 2 x, & if 1 < x \leq 2 \\ 4, & if 2 < x \leq 3 \\ 2 x - 2, & if 3 < x \leq 4 \\ 4 x - 10, & if 4 < x < \infty \end{array}$
Could be shown as
Clearly, the least value of f(x) is 4.
The number of integral solutions of f(x) =4 are two .i .e {2,3}
$\begin{array}{lr} Also, & π - 1, e, \frac{12}{5} \in {2, 3} \\ ∴ & \frac{f (π - 1) + f (e)}{2 f (\frac{12}{5})} = 1 \end{array}$

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