Question-find-the-minimum-value-of-f-x-1-x-2-x-4-2x-

Question Number 157604 by mnjuly1970 last updated on 25/Oct/21

Q u e s t i o n

f i n d {t h e}^{″} {m i n i m u m}^{″} v a l u e o f :

f (x) := ∣ 1 + x ∣ + ∣ 2 + x ∣ + ∣ 4 + 2 x ∣

Commented by mr W last updated on 25/Oct/21

i t h i n k t h e e a s i e s t w a y i s t o c h e c k a l l

k i n k p o s i t i o n s :

x = - 1, - 2

f (- 1) = 0 + 1 + 2 = 3

f (- 2) = 1 + 0 + 0 = 1

\Rightarrow f {(x)}_{m i n} = 1

Commented by mr W last updated on 25/Oct/21

i n t h i s c a s e w e c a n a l s o d o l i k e t h i s :

f (x) =∣ 1 + x ∣ + 3 ∣ 1 + 1 + x ∣=∣ 1 + x ∣ + 3 \pm 3 ∣ 1 + x ∣

= 3 + {\begin{cases} 4 ∣ 1 + x ∣ \\ - 2 ∣ 1 + x ∣ \end{cases}

f {(x)}_{m i n} = 3 - 2 = 1 a t x = - 1

Commented by mnjuly1970 last updated on 25/Oct/21

g r a t e f u l s i r W

Answered by TheSupreme last updated on 25/Oct/21

f (x) =∣ 1 + x ∣ + 3 ∣ 2 + x ∣

f^{'} (x) = s g n (1 + x) + 3 s g n (2 + x) \neq 0 \forall x \in R - {- 1, - 2}

f (- 1) = 3

f (- 2) = 1

l i \underset{x \to \pm \infty}{m} f (x) = \infty

m i n f (x) = 1

Commented by mnjuly1970 last updated on 25/Oct/21

t h a n k s a l o t