Minimum value is -∞ for x in (-∞, -2). Minimum value is 2 for x in (-2, ∞).
It depends on what we are looking at.
y = (x^2 + 2x + 4) / (x+2) = x + 4/(x+2) = -2 + (x+2) + 4/(x+2) **** We can say that z + 4/z is minimum when z = √4 = 2. **** So x+2 = 2, ie., at x = 0, y is minimum. So Minimum value = 2.
==> Alternately: Take derivatives on both sides: dy/dx = 1 - 4 / (x+2)² = 0 for x+2 = +2 or -2 ie., x = 0 or -4 d²y/dx² = 8 /(x+2)³ > 0 for x > -2. Choose x such that y" > 0.
So for x = 0, y is minimum. Minimum value = (0^2 + 2 *0+4)/(0+2) = 2.
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Minimum value is -∞ for x in (-∞, -2).Minimum value is 2 for x in (-2, ∞).
It depends on what we are looking at.
y = (x^2 + 2x + 4) / (x+2)
= x + 4/(x+2) = -2 + (x+2) + 4/(x+2)
**** We can say that z + 4/z is minimum when z = √4 = 2.
**** So x+2 = 2, ie., at x = 0, y is minimum. So Minimum value = 2.
==> Alternately: Take derivatives on both sides:
dy/dx = 1 - 4 / (x+2)² = 0 for x+2 = +2 or -2
ie., x = 0 or -4
d²y/dx² = 8 /(x+2)³ > 0 for x > -2. Choose x such that y" > 0.
So for x = 0, y is minimum.
Minimum value = (0^2 + 2 *0+4)/(0+2) = 2.
See the diagram.