Vipul: Created page with "==Definition== The '''concavity of a function''', or more precisely the '''sense of concavity of a function''', describes the way the derivative of the function is changi..."

2012-06-04T18:07:17Z

Created page with "==Definition== The '''concavity of a function''', or more precisely the '''sense of concavity of a function''', describes the way the derivative of the function is changi..."

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==Definition==

The '''concavity of a function''', or more precisely the '''sense of concavity of a function''', describes the way the [[derivative]] of the function is changing. There are two determinate senses of concavity: ''concave up'' and ''concave down''. Note that it is possible for a function to be neither concave up nor concave down.

===Definition in terms of convex combinations===

A function <math>f</math> defined on an interval <math>I</math> is termed '''concave up''' if it satisfies the following condition: for any <math>x_1,x_2 \in I</math> with <math>x_1 < x_2</math>, and any <math>t \in (0,1)</math>, we have:

<math>f(tx_1 + (1 - t)x_2) < tf(x_1) + (1 - t)f(x_2)</math>

Geometrically, what this means is that the chord joining the points <math>(x_1,f(x_1))</math> and <math>(x_2,f(x_2))</math> in the graph of <math>f</math> lies above the part of the graph of <math>f</math> between <math>x_1</math> and <math>x_2</math>.

A function <math>f</math> defined on an interval <math>I</math> is termed '''concave down''' if it satisfies the following condition: for any <math>x_1,x_2 \in I</math> with <math>x_1 < x_2</math>, and any <math>t \in (0,1)</math>, we have:

<math>f(tx_1 + (1 - t)x_2) > tf(x_1) + (1 - t)f(x_2)</math>

Geometrically, what this means is that the chord joining the points <math>(x_1,f(x_1))</math> and <math>(x_2,f(x_2))</math> in the graph of <math>f</math> lies below the part of the graph of <math>f</math> between <math>x_1</math> and <math>x_2</math>.

===Definition in terms of first derivative===

A function <math>f</math> defined on an interval <math>I</math> is termed '''concave up''' if it satisfies the following conditions:

# The derivative of <math>f</math> is defined everywhere on <math>I</math> except at possibly a discrete set of points.
# Consider each of the intervals where the derivative of <math>f</math> is defined. The derivative of <math>f</math> is an increasing function on that interval.
# At each of the points in the interior of the domain where the derivative of <math>f</math> is not defined, the following are true: <math>f</math> is continuous, <math>f</math> is left differentiable, <math>f</math> is right differentiable, and the left hand derivative of <math>f</math> is less than the right hand derivative of <math>f</math>.
# At the endpoints of the interval, if they exist, <math>f</math> is appropriately one-sided continuous and one-sided differentiable.

A function <math>f</math> defined on an interval <math>I</math> is termed '''concave down''' if it satisfies the following conditions:

# The derivative of <math>f</math> is defined everywhere on <math>I</math> except at possibly a discrete set of points.
# Consider each of the intervals where the derivative of <math>f</math> is defined. The derivative of <math>f</math> is a decreasing function on that interval.
# At each of the points in the interior of the domain where the derivative of <math>f</math> is not defined, the following are true: <math>f</math> is continuous, <math>f</math> is left differentiable, <math>f</math> is right differentiable, and the left hand derivative of <math>f</math> is greater than the right hand derivative of <math>f</math>.
# At the endpoints of the interval, if they exist, <math>f</math> is appropriately one-sided continuous and one-sided differentiable.

Concavity of a function - Revision history

Vipul: Created page with "==Definition== The '''concavity of a function''', or more precisely the '''sense of concavity of a function''', describes the way the derivative of the function is changi..."