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renyikouniao
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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 $$\frac1{b-a}\int_a^bf(x)\,dx > \tfrac12(f(a)+f(b)).$$renyikouniao said: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.
A concave downward function is a mathematical function that curves downward, forming a concave shape. In other words, the function decreases at an increasing rate as the input values increase.
A convex function curves upward, while a concave downward function curves downward. In a convex function, the slope is increasing as the input values increase, while in a concave downward function, the slope is decreasing as the input values increase.
The average of a concave downward function represents the average rate of change of the function over a given interval. This can also be interpreted as the slope of the line connecting the endpoints of the interval.
The average of a concave downward function can be calculated by finding the average rate of change of the function over a given interval. This can be done by finding the slope of the secant line connecting the endpoints of the interval, or by finding the average of the slopes of multiple tangent lines at different points within the interval.
Concave downward functions and their averages are commonly used in economics, finance, and engineering. For example, the demand for a product can often be modeled using a concave downward function, and the average rate of change of this function can represent the elasticity of demand. In finance, concave downward functions and their averages are used to model stock prices and interest rates. In engineering, they can be used to analyze the speed and acceleration of a moving object.