First slide

Maxima and minima

Question

$The minimum value of f (x) = | 3 - x | + | 2 + x | + | 5 - x | is$

Moderate

A

0
B

7
C

8
D

10

Solution

$\begin{array}{l} f can be written as \\ \begin{aligned} f (x) & = \{\begin{cases} 3 - x - (2 + x) + 5 - x & if x \leq - 2 \\ 3 - x + 2 + x + 5 - x & if & - 2 < x < 3 \\ x - 3 + 2 + x + 5 - x & if 3 \leq x < 5 \\ x - 3 + 2 + x + x - 5 & if x \geq 5 \end{cases} \\ = \{\begin{matrix} 6 - 3 x if x \leq - 2 \\ 10 - x if - 2 < x < 3 \\ x + 4 if 3 \leq x < 5 \\ 3 x - 6 if x \geq 5 \end{matrix} \end{aligned} \end{array}$
By drawing the graph of f(x) we see that minimum value of f is 7.

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