A function $f:\left [ a,b \right ]\rightarrow \mathbb{R} $ is called convex in the interval $I= \left [ a,b \right ]$, if for any $t\in \left [ 0,1 \right ] $ and for all $a\leq x< y\leq b$ the following inequality holds
$f\left ( ty+\left ( 1-t \right ) x\right )\leq tf\left ( y \right )+ \left ( 1-t \right )f\left ( x \right )$
convex function (bomc)
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nafistiham
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\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
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Re: convex function (bomc)
Basically, $ty+\left ( 1-t \right ) x$ is the line between $x$ and and $y$ and $tf\left ( y \right )+ \left ( 1-t \right )f\left ( x \right )$ is the line between $f(x)$ and $f(y)$. So, the inequality means that for any point $z$ between $x$ and $y$, $f(z)$ is under the line between $f(x)$ and $f(y)$.
Re: convex function (bomc)
Look at the picture in the book. There is a line between $f(x)$ and $f(y)$. The value of $f(z)$ for any $z$ between $x$ and $y$ is below that line. Do you see how the definition is just saying this?
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nafistiham
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Re: convex function (bomc)
oops,got it. 
actually i was confused about the function part.thanks a lot. 
actually i was confused about the function part.thanks a lot. \[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.
Re: convex function (bomc)
And a concave function is the reverse-
A function $f:\left [ a,b \right ]\rightarrow \mathbb{R} $ is called concave in the interval $I= \left [ a,b \right ]$, if for any $t\in \left [ 0,1 \right ] $ and for all $a\leq x< y\leq b$ the following inequality holds
$f\left ( ty+\left ( 1-t \right ) x\right )\geq tf\left ( y \right )+ \left ( 1-t \right )f\left ( x \right )$
A function $f:\left [ a,b \right ]\rightarrow \mathbb{R} $ is called concave in the interval $I= \left [ a,b \right ]$, if for any $t\in \left [ 0,1 \right ] $ and for all $a\leq x< y\leq b$ the following inequality holds
$f\left ( ty+\left ( 1-t \right ) x\right )\geq tf\left ( y \right )+ \left ( 1-t \right )f\left ( x \right )$
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Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi