MHB Concave Downward Function and Average

[renyikouniao]

Please show that if f is a continuous concave downward function on [a,b],then the average of the function f is greater than f[(a+b)/2],thank you in advance.

 

[Ackbach]

This conclusion is false. Consider $-x^{2}+1$ on the interval $[-1,1]$. It's continuous and concave downward everywhere. Then the average value of the function is
$$ \frac{1}{1-(-1)} \int_{-1}^{1}(-x^{2}+1) \, dx= \int_{0}^{1}(-x^{2}+1) \, dx
= \frac{2}{3},$$
but
$$f\left( \frac{1+(-1)}{2}\right)= f(0) = 1 > \frac{2}{3} .$$

 

[renyikouniao]

awesome explanation!thank you.

 

[Opalg]

Please show that if f is a continuous concave downward function on [a,b],then the average of the function f is greater than f[(a+b)/2],thank you in advance.

Perhaps the question should have read "Show that if f is a continuous concave downward function on [a,b],then the average of the function f is greater than [f(a)+f(b)]/2]." That is true, and you see why geometrically, if you notice that the graph of f lies above the line joining the points (a,f(a)) and (b,f(b)). Thus the area under the graph is greater than the area of the trapezium with vertices (a,0), (a,f(a)), (b,f(b)), (b,0). In other words, $$\int_a^bf(x)\,dx > \tfrac12(f(a)+f(b)(b-a),$$ from which $$\frac1{b-a}\int_a^bf(x)\,dx > \tfrac12(f(a)+f(b)).$$