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I would like to show that this function

$$2x^2 + 8y^2$$

is convex or concave by using the definition

$$f(θx+(1−θ)y) \le θf(x)+(1−θ)f(y)$$

From my understanding, using the Hessian matrix, I believe for certain I can show that the function is positive definite and hence convex.

But I'm not certain how to proceed on showing this using the equation shown above.

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Your function is a function of two variables so you have: $$f(x,y)= 2x^2 + 8y^2 \ \ \forall(x,y)$$ So in the definition, you need to show that for two tuples of variables like $z_1=(x_1, y_1)$ and $z_2=(x_2,y_2)$ the definition holds. Specifically, you can show the following, which need some algebraic calculations.

$$f(\theta z_1+(1-\theta)z_2)\le\theta f(z_1)+(1-\theta)f(z_2)$$

For $\theta \in[0,1]$ convexity and for $\theta \in(0,1)$ and $z_1 \neq z_2$ strict convexity can be shown. See also this question in math SE.

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The sum of convex functions is convex!

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