Jensen s Inequality Versus Algebra. in Finding the Exact Maximum

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1 International Mathematical Forum, Vol. 6, 0, no. 59, Jensen s Inequality Versus Algebra in Finding the Exact Maximum Mohammad K. Azarian Department of Mathematics University of Evansville 800 Lincoln Avenue, Evansville, IN 477, USA azarian@evansville.edu Abstract In this article we find the exact maximum value of a function without the conventional method of using critical numbers. In fact, we find the exact maximum without even finding the derivative of the function. First we apply Jensen s Inequality, and then we use simple algebra to find the exact maximum. We conclude the article by posing two questions for the reader. Mathematics Subject Classification: 6A, 6D Keywords: concave function, convex function, Jensen s Inequality. Preliminaries When dealing with the extrema of a function the first thing that comes to mind is to calculate the derivative of the function and find the critical numbers (provided they exist). Sometimes, finding the exact derivative and/or

2 950 M. K. Azarian the exact critical numbers of a function could be very challenging if not impossible. Our goal in this article is to find the exact maximum value of a function without the conventional method of finding and using critical numbers. First we apply Jensen s Inequality, and then we use simple algebra to find the exact maximum. It is intriguing to find the exact maximum value of a rather peculiar function without finding its critical numbers or even being concerned about its derivative. As the reader may know there are many versions of Jensen s Inequality in the real and complex analysis of one or several variables. Various versions of Jensen s Inequality is used in probability theory, measure theory, the theory of integration, and trigonometry, just to name a few. In most cases Jensen s Inequality is applicable only if the function under discussion is convex or concave. In this article, we need the most common form of Jensen s Inequality, namely the following theorem: Theorem.. (Jensen s Inequality). A continuous real-valued function f(x) defined on an interval I R is concave if and only if for any x,x I. f(x )+f(x ) f( x + x ), We note that (i) the above theorem can be generalized for f(x,x,..., x n ), and (ii) the direction in the above inequality changes if and only if f(x) is convex.. The Result In this section first we state the problem, and then we apply Jensen s Inequality as well as simple algebra to find the exact maximum... The Problem

3 Jensen s inequality versus algebra 95 The maximum value of f(α) = ( + sin α cos α) [( + cos α) + ( + sin α) ] is exactly, provided 0 α π. We note that the derivative of f is a rather long expression, and one cannot find the exact critical point(s) and hence the exact maximum of f, by a conventional method. Therefore, to find the exact maximum of f, first we make use of Jensen s Inequality and then we just use elementary algebra as follows... Using Jensen s Inequality We need to show that f(α) = ( + sin α cos α) [( + cos α) + ( + sin α) ], for α in [0, π ]. Equivalently, we need to show that + cos α + + sin α + sin α cos α, () for α in [0, π ]. Now, we note that if α =0orα = π, then in both cases we have +, which is a true statement. Therefore, we may assume that α is in (0, π ), and hence cos α>0 and sin α>0. Next, since both cos α and sin α are positive in (0, π ), we choose nonnegative real numbers r and s such that cos α = e r and sin α = e s. Substituting cos α = e r and sin α = e s in the Inequality (), we obtain +, () +e r +e s +e (r+s) for r, s 0. Now, we consider the function f(x) = +e x.

4 95 M. K. Azarian To prove the Inequality (), by Jensen s Inequality we need to show that f(r)+f(s) f( r + s ), (3) for r, s 0. f is concave. By Jensen s Inequality, the Inequality (3) is true if and only if But, since f e x (x) = < 0, e 4x ( + e x ) 5 for all x 0, f is concave, as desired. Finally, we note that we have equality when α = π, and therefore, the maximum value of f(α) is exactly Using Basic Algebra If t [0, ], then it is easy to see that (t 3t +5) 4( t +) ( t 4 +). (4) Now, if we divide both sides of the Inequality (4) by ( t +), and then take the square roots of both sides we get t 3t +5 t + = 3+ t +8 t t + 4 +, which is equivalent to t +8 t (5) + 4 t Next, if we let t = sin α cos α, then the Inequality (5) can be rewritten as 4 sin α cos α +8 sin α cos α + 3+ sin α cos α +, which is equivalent to ( + sin α cos α ) ( + sin α + + cos α). (6) + sin α cos α But, the Inequality (6) can be rewritten as sin α cos α + = ( + sin α)( + cos α) + sin α + + cos α, + sin α cos α + sin α cos α

5 Jensen s inequality versus algebra 953 which is equivalent to + sin α cos α Hence, from the Inequality (7) we deduce that + cos α + + sin α. (7) ( + sin α cos α) [( + cos α) + ( + sin α) ]. Moreover, we have equality when t = sin α cos α = sin α =. when α = π. Therefore, the maximum value of f(α) is exactly. 4 That is, 3. Questions Question 3.. Can you think of other functions where their exact maximum (minimum) can be found using Jensen s Inequality as well as just simple algebra? Question 3.. Can the above problem be generalized for f(α,α,..., α n ) using both generalized Jensen s Inequality and simple algebra? References [] M. K. Azarian, On the Fixed Points of a Function and the Fixed Points of its Composite Functions, International Journal of Pure and Applied Mathematics, Vol. 46, No., 008, pp Mathematical Reviews, MR43373 (009c:659), March 009. Zentralblatt MATH, Zbl [] M. K. Azarian, There May be More Than One Way to Find the Derivative of a Function, Missouri Journal of Mathematical Sciences, Vol. 5, No., Winter 993, pp Zentralblatt MATH, Zbl [3] G. H. Hardy, J. E., Littlewood, G. Polya, Inequalities, Second Edition, Cambridge University Press, Cambridge, England, 988.

6 954 M. K. Azarian [4] I. Niven, Maxima and Minima Without Calculus, The Mathematical Association of America, Washington, D. C., 98. Received: June, 0