The geometric, logarithmic, and arithmetic mean inequality a. Show that the graph of e^x i

step 1 So this first question has three parts. The first question asks us to show that the graph of E t

The geometric, logarithmic, and arithmetic mean inequality ...

The geometric, logarithmic, and arithmetic mean inequality \begin{equation}\begin{array}{l} \quad \quad \quad\quad\quad{\text { a. Show that the graph of } e^{x} \text { is concave up over every interval }} \\ \quad \quad\quad\quad\quad\quad{\text { of } x \text { -values. }} \\ \quad\quad\quad\quad\quad{\text { b. Show, by reference to the accompanying figure, that if }} \\ \quad\quad\quad\quad\quad\quad{0 < a < b, \text { then }}\end{array}\end{equation} \begin{equation}e^{(\ln a+\ln b) / 2} \cdot(\ln b-\ln a)<\int_{\ln a}^{\ln b} e^{x} d x<\frac{e^{\ln a}+e^{\ln b}}{2} \cdot(\ln b-\ln a).\end{equation} \begin{equation}\begin{array}{c}{\text { c. Use the inequality in part (b) to conclude that }}\end{array}\end{equation} \begin{equation}\sqrt{a b}<\frac{b-a}{\ln b-\ln a}<\frac{a+b}{2}.\end{equation} \begin{equation}\begin{array}{l} \quad\quad \quad\quad \quad\quad {\text { This inequality says that the geometric mean of two positive }} \\ \quad\quad \quad\quad \quad\quad {\text { numbers is less than their logarithmic mean, which in turn is }} \\ \quad\quad \quad\quad \quad\quad {\text { less than their arithmetic mean. }}\end{array}\end{equation}

The geometric, logarithmic, and arithmetic mean inequality
\begin{equation}\begin{array}{l} \quad \quad \quad\quad\quad{\text { a. Show that the graph of } e^{x} \text { is concave up over every interval }} \\ \quad \quad\quad\quad\quad\quad{\text { of } x \text { -values. }} \\ \quad\quad\quad\quad\quad{\text { b. Show, by reference to the accompanying figure, that if }} \\ \quad\quad\quad\quad\quad\quad{0 < a < b, \text { then }}\end{array}\end{equation}
\begin{equation}e^{(\ln a+\ln b) / 2} \cdot(\ln b-\ln a)<\int_{\ln a}^{\ln b} e^{x} d x<\frac{e^{\ln a}+e^{\ln b}}{2} \cdot(\ln b-\ln a).\end{equation}
\begin{equation}\begin{array}{c}{\text { c. Use the inequality in part (b) to conclude that }}\end{array}\end{equation}
\begin{equation}\sqrt{a b}<\frac{b-a}{\ln b-\ln a}<\frac{a+b}{2}.\end{equation}
\begin{equation}\begin{array}{l} \quad\quad \quad\quad \quad\quad {\text { This inequality says that the geometric mean of two positive }} \\ \quad\quad \quad\quad \quad\quad {\text { numbers is less than their logarithmic mean, which in turn is }} \\ \quad\quad \quad\quad \quad\quad {\text { less than their arithmetic mean. }}\end{array}\end{equation}

Key Concepts

Concavity and Convexity

This concept involves studying the curvature of a function’s graph. A function is concave up (or convex) on an interval if its second derivative is positive throughout that interval. This property is crucial for understanding how the function behaves relative to its chords and tangents, and it is a key tool in proving various inequalities using the geometric interpretation of integrals.

Exponential Function Properties

The exponential function, particularly e^x, has unique qualities such as the rate of change being equal to the function’s value and its graph exhibiting continuous growth. Recognizing that e^x is convex (or concave up) allows one to apply concepts from calculus like the second derivative test and to use the function's behavior in deriving inequalities involving means.

Definite Integration and Area Under a Curve

Definite integrals compute the area under the curve of a function between specified limits. This concept is utilized to compare the actual area under a convex function with approximations derived from linear (or chord) approximations. In the context of inequalities, integration serves as a bridge between the average values of a function and its geometric interpretation.

Mean Value Theorem and Average Value

The Mean Value Theorem guarantees that for continuous functions, there exists a point where the derivative (or the function itself) equals the average rate of change or average value over an interval. This theorem underpins the transition from integral expressions of a function to its average value, thereby providing a pathway to establish relations among different types of means.

Inequalities Among Means

This concept involves comparing different forms of averages—specifically the geometric, logarithmic, and arithmetic means. Such inequalities are fundamental in mathematics to understand how various averaging processes relate to one another, often revealing deeper properties about the numbers or functions involved. Deriving these inequalities typically requires employing properties from calculus, such as the convexity of functions and the use of integrals.

Transcript

00:01 So this first question has three parts.

00:02 The first question asks us to show that the graph of e the x is concave up over all x values.

00:10 So we have y is equals e to the x.

00:14 The point thing to remember is we can see concavity with the second derivative of the function.

00:20 So first i'm going to find the second derivative.

00:22 So y prime.

00:24 Deriv of e to the x is just e to the x times the derivative of the exponent, which here is just one.

00:32 So second derivative is just exactly e to the x.

00:38 And something we know about e to the x is that it is always greater than zero for all x.

00:47 So therefore that implies that the graph of e to the x is always concave upward.

01:05 So that's the answer for part a.

01:08 For part b, using this figure, we're supposed to show that this long inequality is, true.

01:16 So first looking at this picture, just to see what i noticed.

01:20 I notice i have these two kind of trapezoids here.

01:28 I have this first one, and then i have another bigger one here.

01:36 And then i have my function.

01:40 So i'm going to be comparing the area between this trapezoid, this bigger one, and the area underneath the curve.

01:48 So first, what i notice is that this area of this trapezoid, so the area of trapezoid, what is this, a, b, c, d.

02:08 I notice that this area is slightly smaller than the area of underneath the curve y is equal to e to the x, which defined area.

02:17 Again, it's just the integral of it, which this is going from natural log of a to natural log of b.

02:31 And then i noticed that the area of this bigger trapezoid contains more in it.

02:37 It's a bigger area than underneath this curve.

02:40 So this is going to be less than the area trapezoid aefd.

02:51 So i have this inequality.

02:54 So now i just want to find out what my areas of these trapezoids are.

02:59 So an important thing to remember is that the area of a trapezoid is one -half, face one, plus, base 2 times height.

03:13 So looking at my abcd, i see that my height here is the straight line here.

03:27 So it's going to be the natural log of b minus the natural log of a.

03:35 So on this left hand side i have one half times the height natural log of b minus natural log of a times base 1 plus base 2.

03:46 In here i don't really know what those values are i just know it's a b plus cd so it's gonna be less than my integral here and now looking at the area of the other trapezoid i still have this same height natural log of b minus natural log of a however this one i do know actually where my bases are i see that these go until they hit my function here so these heights or the length of these bases are actually just going to be these x values at into my function.

04:38 So here this point is natural log of a, e to the natural log of a...

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