Understanding Concavity

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Hello friends,
I'm trying to understand Concavity, But I've some doubts. As we know quadratic polynomial on graphing forms a parabola opening upwards or downwards. While understanding concavity we divide graphs into separate region using inflection points and each separate region is a quadratic polynomial in its own. So, its derivative should be linear. Then why the derivative of function ' f ' in the domain (vertical green line) is not linear.

http://s11.postimg.org/x23anpmsj/Capture.png

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Type: Question • Score: 4 • Views: 1,370 • Replies: 9

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View best answer, chosen by 22990atinesh

@22990atinesh,

Irrespective of concavity considerations, the derivative of a function f will only be linear if the function is itself a quadratic. The function (blue) here is clearly not a quadratic.
To determine maxima, minima, or points of inflection, the second derivative f '' is tested for negative, positive, or zero values (respectively) at the turning point.

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@fresco,

Hello fresco, how u are saying the function f is not quadratic (bounded by green vertical lines). It is clearly visible its a parabola so its derivative must be linear. Any curve appearing like a parabola is quadratic.

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@22990atinesh,

No. You simply cannot LOOK at a section of a function and call it "a parabola" ! The total blue trace clearly does not conform to ax^2 +bx +c. The number of turning points visible implies it is at least a quartic (a^4 etc)
If you want to prove concavity within the stated domain, you need to show that the second derivative does not change its sign within the domain, and that a maximum or minimum lies within the domain.

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@22990atinesh,

NB Note also that if the (blue) function were parabolic in the domain the (orange) first derivative would be a straight line within the domain.

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Quote:

Concavity Theorem: If the function f is twice differentiable at x=c, then the graph of f is concave upward at (c,f(c)) if f"(c)>0 and concave downward if f"(c)<0.

Rap

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@fresco,

Hello fresco,
I understood my misconception about the quadratic curves. Quadratic curves are parabola which has to be symmetric about its "Axis of Symmetry". And as the graph under domain is not symmetric, although it is curved but it is not a quadratic curve right.

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@22990atinesh,

Right. But you have to classify the whole curve which is not of a quadratic.

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@22990atinesh,

Just to be clear, it is not enough to look parabolic and be symmetric to be a parabola. Parabolas are second degree equations.

y = x^4 is symmetric and has a parabolic look to it, but it is not a parabola.

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Some are feeling convexedity over concavity.

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Understanding Concavity

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