Is $f_a$ convex function for various value of $a$?

Asked Jun 21 '17 at 08:11

Active Jun 21 '17 at 09:24

Viewed 36 times

$f_{a}:[a,1]\rightarrow \mathbb{R}$, where $a\leq 0$

$$f_{a}(x)=\begin{cases} 1 & \text{ if } x=0 \\ 0& \text{ if } x \neq 0 \end{cases}$$ A) $f_{a}$ is convex for all $a\leq 0$

B) $f_{a}$ is not convex for any $a\leq 0$

C) $f_{a}$ is a convex function for all $a< 0$ and not convex for $a = 0$

D) $f_{a}$ is not convex function for any$a< 0$ and convex for $a =0$

This is what the ploted graph looks like right?

How do we know id it's convex?

edited Jun 21 '17 at 09:24

asked Jun 21 '17 at 08:11

user405925

what does $\neq > $ and $= > $ mean? do they mean $\neq $ and $ \geq$ respectively? – Siong Thye Goh Jun 21 '17 at 08:16
1

@SiongThyeGoh Now go ahead.. – Red shoes Jun 21 '17 at 08:34
what have you tried? are there some values for $a$ where you already have a solution? – supinf Jun 21 '17 at 08:46
Well,it's linear everywhere except at a=0 it is a single point. I don't know what conclusion we can draw from this? – user405925 Jun 21 '17 at 09:05
In the definition of $f_a$, @MAthsjunky you said $a\leq 0$, in statement (C),(D) you have $a\geq 0$, $a=0$ would be better. – MAN-MADE Jun 21 '17 at 09:13
Yes, i fixed it. That was the original question, someone edited it. – user405925 Jun 21 '17 at 09:24
Can you state the solution ,please? – user405925 Jun 21 '17 at 09:25
what happens if you apply the definition of convexity? – supinf Jun 21 '17 at 09:34
@MAthsjunky I think you need to use Jensen's inequality – MAN-MADE Jun 21 '17 at 10:06

Is $f_a$ convex function for various value of $a$?

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