What is the extreme |x|+|x+1|+|x^2+1| ?

The extreme |x|+|x+1|+|x^2+1| is Minimum(0,2)

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extreme |x|+|x+1|+|x^2+1|

extreme points

∣ x ∣ + ∣ x + 1 ∣ + x^{2} + 1

Minimum (0, 2)

Solution steps

Find the critical points:

x = - 1, x = 0

Find intervals:

Decreasing

: - \infty < x < - 1,

Decreasing

: - 1 < x < 0,

Increasing

: 0 < x < \infty

Plug the extreme point

x = 0

into

∣ x ∣ + ∣ x + 1 ∣ + x^{2} + 1 \Rightarrow y = 2

Minimum (0, 2)

y = x z

2 cos^{2} (x)

f (x, y) = 4 x^{2} + y^{2} + 1

f (x) = 5 e^{x^{2} - 10 x}

f (x) = 2 x^{2} (x^{2} - 2)

What is the extreme |x|+|x+1|+|x^2+1| ?
The extreme |x|+|x+1|+|x^2+1| is Minimum(0,2)