Exponents and Logarithms, inequalities, area, trapezoid, y = 1/xThe arithmetic-logarithmic-geometric mean inequality states that if 0<a<b then sqrt(ab) < (b-a)/(lnb – lna) < (a+b)/2 Left graphic: The area under y = 1/x on the interval (a,b) is lnb - lna The area under the tangent at x = (a+b)/2 is 2(b-a)/(a+b) Then lnb – lna > 2(b-a)/(a+b) Right graphic: The area under y = 1/x on the interval (a,b) is lnb - lna, as in the left graphic. The area of the left trapezoid is ½(1/a + 1/(sqrt(ab)))(sqrt(ab) - a) = (b-a)/2sqrt(ab) The area of the right trapezoid is ½(1/b + 1/(sqrt(ab)))(b - sqrt(ab)) = (b-a)/2sqrt(ab) Then lnb – lna < (b-a)/sqrt(ab)Componente Curricular::Educação Superior::Ciências Exatas e da Terra::Matemática