Solving Exercise - Proving f(1 + w) > f(1) + w*f(1)

[IlanSherer]

Hello :)

Help please about the following exercise :

The function f(x) comes up and is an convex which gets out from (0,0) and continues into the first quarter.
Prove that for all w > 0, takes place f(1 + w) > f(1) + w*f(1).

I have been trying for two days to solve it :\

Thanks a lot!

 

[I like Serena]

Hello :)

Help please about the following exercise :

The function f(x) comes up and is an convex which gets out from (0,0) and continues into the first quarter.
Prove that for all w > 0, takes place f(1 + w) > f(1) + w*f(1).

I have been trying for two days to solve it :\

Thanks a lot!

Hi IlanSherer! Welcome to MHB! (Smile)

A function is defined to be convex if the line segment between any two points on the graph of the function lies above or on the graph.

Let's pick the points $(0,0)$ and $(1,f(1))$.
Then any point on the corresponding line segment is above the graph, and any point on the line to the right, is below the graph.
And that line is given by $y=f(1) x$.
Yes?

It looks like this:
\begin{tikzpicture}[xscale=3]
\def\x{3};
\draw[<->] (0,0) -- (3.3,0) node

{$x$};
\draw[<->] (0,0) -- (0,9.5) node[above] {$y$};
\draw[help lines] (1,-0.1) node[below] {1} -- (1,1) -- (-0.03,1) node

{f(1)};
\draw[help lines] (\x,\x) -- (-0.03,\x) node

{f(1)+wf(1)};
\draw[help lines] (\x,-0.1) node[below] {1+w} -- (\x,{\x^2}) -- (-0.03,{\x^2}) node

{f(1+w)};
\draw[domain=-0:3.1, variable=\x, red, ultra thick] plot ({\x}, {(\x)^2}) node

{y=f(x)};
\draw[blue, ultra thick] (0,0) -- (3.2,3.2) node

{y=f(1)x};
\draw[blue, ultra thick] (0,0) -- (1,0) -- node

{f(1)} (1,1) -- node[above] {w} (\x,1) -- node

{wf(1)} (\x,\x);
\end{tikzpicture}

So for $w \ge 0$, we have:
$$f(1+w) \ge f(1)(1+w) = f(1) + wf(1)$$

And if the function is strictly convex, we have for $w>0$:
$$f(1+w) > f(1) + wf(1)$$​