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$$f[lambda x_1+(1-lambda)x_2]leqlambda f(x_1)+(1-lambda)f(x_2)quadforall space 0 < lambda < 1$$

How do the coefficients $lambda$ and $1- lambda$ satisfy the convexity of $f$?

The right hand side is a parameterisation of the straight line between $f(x_1)$ and $f(x_2)$. The left hand side is the point on the function with the same $x$-value as the point on the straight line on the right hand side. So this says that the straight line between any two points lies entirely above the function. Equivalently, it says that ${(x,y)|y geq f(x)}$ is a convex set, in the usual sense.

The idea is that the value of $f$ at a point between $x_1$ and $x_2$ is less that the value at the same point for the line segment between $f(x_1)$ and $f(x_2)$.

(credit Wikipedia)

The expression $lambda x_1+(1-lambda)x_2$ is just a parametrization for all the points between $x_1$ and $x_2$ and $lambda f(x_1)+(1-lambda)f(x_2)$ is the parametrization for the line segment between $f(x_1)$ and $f(x_2)$.

The concept can be generalized for more points by Jensen's inequality.

You can see a convex curve as "always turning left", so that it cannot meet a straight line more than twice.

Your equation describes the curve and a chord between two points, and expresses that they do not intersect.

The intuition is that when a function is "really" convex, for each two points $(x_,f(x))$ and $(y,f(y))$ the corresponding connecting line segment falls upper than the function between those two points which is a direct intuition of convexity . $0<lambda<1$ means in fact the middle of the two points.