DSE 2011 Q.24

[Chinni18]

If a function F(x)= Integral a to b f(t)dt is increasing on an interval, then on that interval F is
convex
concave
increasing
decreasing
 Answer key says convex
But consider the following figure, where a convex combination of points related to b and c lie outside the set.
Shouldn't the answer be increasing?
I don't understand, I might be wrong, could someone shed some light on how they did this?

[Khagesh]

Even i think the ans is wrong n shld b increasing.......


For reference read condition here:

http://tutorial.math.lamar.edu/pdf/Calculus_Cheat_Sheet_Derivatives.pdf

[Chinni18]

Ritu, the function f(t) in indeed convex, but the function F(x) is definitely not, because it is a closed set but doesn't satisfy the condition of convexity

[Chinni18]

Thanks Khagesh, that sheet clarified things. Maybe the answer key has an error.

[ritu]

if i m nt wrong ur thinking abt "convex set" rather than a "convex function"...the graph u have shown is depicting a convex function..

[ritu]

ok..i dnt kno then....sorry...

[Chinni18]

Hmm. I see what you mean Ritu... My only doubt is that f(t) is a convex function all right, I agree. But are we supposed to look at F(x) in the sense of a set or a function?

[duck]

Hi Chinni.. :)

F(x) is a function not a set.
And the correct answer is Convex only.
Option(c) i.e "increasin" can be ruled out using the following example>
Consider f(t) = -2. Its an increasing and convex function but F(x) = -2x which is a decreasing function.

[Chinni18]

Ohhhh.... Ya you're right!!!!! Thanks duck!!!!! You're a life-saver!!!

On Thu, Jun 21, 2012 at 6:29 PM, duck [via Discussion forum] <[hidden email]> wrote:

Hi Chinni.. :)

F(x) is a function not a set.
And the correct answer is Convex only.
Option(c) i.e "increasin" can be ruled out using the following example>
Consider f(t) = -2. Its an increasing and convex function but F(x) = -2x which is a decreasing function.

:)

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[AJ]

@duck.

I thought the answer convex for this question by the logic like we are given that f'(x)>=0 .. this is second derivative of integral of f(x) .. so F''(x) >= 0 .. So, convex.

:p

Is there any flaw in this reasoning ?? I am sure I am making some stupid mistake here....
pls reply.

[AJ]

Please someone reply to this.....