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Convex Function and Concave Function
Motivation:
In previous section, the method to determine if a function is increasing or decreasing in an interval has been introduced. Other than increasing or decreasing, we would like to know how it increases or decreases. For example, both of the following two diagrams are increasing functions.
Obviously, they are different : one is like an upward mouth, and the other is like a downward mouth. But mathematically, how can we express the difference ? Naturally, we would compare it with a line by connecting two points on the curve and check if the curve is “above” the line ( secant line ) or “below” the line between the two points.
Of course, whether the “mouth” of the curve is upward or downward is nothing to do with that it is increasing or decreasing. It is just another characteristic of the curve.
And recall that the result for the coordinate of a point in a line segment.
If C is on , and : = (1-p) : p , then
= p + (1-p)
Thus, we can use this result to express the concept for “above the line” or “below the line” as indicated previously.
And please also notice the change of the slopes of tangent lines: the change of the slope is monotonically decreasing or increasing. So, if the derivative function of the curve exists, we might have the chance to know it is convex or concave from the derivative function.
Definition ( Convex and Concave ) 0 p 1 . Let f be a function defined in the interval [a,b] . If
pf(c)+(1-p)f(d) f(pc+(1-p)d) for any subinterval [c,d] in [a,b]
then we say that f is concave in the interval [a,b] . Similarly, if
pf(c)+(1-p)f(d) f(pc+(1-p)d) for any subinterval [c,d] in [a,b]
then we say that f is convex in the interval [a,b] .
Theorem : f is convex f(d) f(c) + f’(c)(d-c) for any c, d in the domain of f .
Proof: ( ) f is convex So, p f(d) + (1-p) f(c) f( pd + (1-p)c ) for 0 < p < 1 .
p( f(d) – f(c) ) + f(c) f( p(d-c) + c) p( f(d) – f(c) ) f( p(d-c) + c) – f(c) f(d) – f(c) f(d) – f(c)
As p 0 , we have f(d) – f(c) f’(c)(d-c)
Thus, f(d) f(c) + f’(c)(d-c) for any c, d .
( ) f(d) f(c) + f’(c)(d-c) for any c, d . So, for any a, b in the domain of f , we have
t = pa+(1-p)b and f(a) f(t) + f’(t)(a-t) f(b) f(t) + f’(t)(b-t)
Hence, p f(a) + (1-p) f(b) p(f(t) + f’(t)(a-t) ) + (1-p)( f(t) + f’(t)(b-t) ) p f(a) + (1-p) f(b) f(t) + f’(t)( (1-p)(b-t) + p(a-t) ) p f(a) + (1-p) f(b) f(t) + f’(t)( (1-p)b + pa – t )
but t = pa+(1-p)b , we have
p f(a) + (1-p) f(b) f(t)
Basically, the theorem is about the relationship between the slope of the line by choosing two points on the function and the slope of the tangent line for a convex function. In other words,
for convex function f
It also implies the observation on the change of the slopes of the tangent lines at the beginning of this section for convex function is just equivalent to the way we define a convex function. And the similar property also exists for “concave function”. For a concave function g, -g will be convex. And we use the result above directly. If we apply the result above on various points in an interval, we can just state the following properties without rigorous proof.
Properties: Let f(x) be a function such that its derivative exists. Then (1) when f’(x) is monotonically increasing in an interval [a,b], f(x) is convex in this interval. (2) when f’(x) is monotonically decreasing in an interval [a,b], f(x) is concave in this interval.
To check if f’(x) is monotonically increasing or decreasing, naturally we will think about to find its derivative is larger or smaller than 0 . The derivative of f’(x) is known as 2nd order derivative of f(x), denoted as f”(x).
Theorem ( 2nd order derivative ) Let f(x) be a function such that its derivative and 2nd order derivative exist. If f”(c) > 0, then f(x) is convex in the neighborhood of x=c; if f”(c) < 0, then f(x) is concave in the neighborhood of x=c.
Example: f(x)=x3 -2x2 -4x+1
f’(x) = 3x2 – 4x -4 = (3x+2)(x-2) f”(x) = 6x – 4
Thus, we can conclude as follows:
So, f(x) roughly looks like
To gain some confidence, the plot generated from computer is as follows
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