Is the Quadratic function $ f \left( x_1, x_2, x_3 \right) $ Convex?

Question

Given the Quadratic Function $ f \left( x_1,x_2,x_3 \right) $:

$$ f \left( x_1,x_2,x_3 \right) = 2x_1^2 - 2x_1x_2 + x_2^2 + x_3^2 + 2x_1 - x_2 + 2x_3 - 3 $$

Is $ f \left( x_1,x_2,x_3 \right) $ convex?

Answer 1

See that:

$$f(x_1,x_2,x_3)=(x_1-x_2)^2+(x_1+1)^2+(x_3+1)^2-x_2-5$$

and

$$f_1(x_1,x_2,x_3)=(x_1-x_2)^2$$

$$f_2(x_1,x_2,x_3)=(x_1+1)^2$$

$$f_3(x_1,x_2,x_3)=(x_3+1)^2$$

$$f_4(x_1,x_2,x_3)=-x_2-5$$

are all convex functions.

Write

$$f=f_1+f_2+f_3+f_4$$

and apply the definition of convex function for each $f_i$.