Problem 43 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(f^{\prime \prime}(x) \geq 0 \forall x \in[a, b]\), show that \(f\left(\frac{x_{1}+x_{2}}{2}\right) \leq \frac{f\left(x_{1}\right)+f\left(x_{2}\right)}{2}\) for \(x_{1}, x_{2} \in[a, b]\).

Step-by-Step Solution

Answer

The statement \(f(\frac{x_1 + x_2}{2}) \leq \frac{f(x_1) + f(x_2)}{2}\) for \(x_1, x_2 \in [a, b]\) is proved to be true for a function \(f(x)\) that is convex over the interval [a, b]. This is achieved by first understanding and applying Kepler's Inequality and then using the given second derivative condition to infer that the function is convex over the specified interval.

Step 1 - Understanding Kepler's Inequality

Consider two points, \(x_{1}\) and \(x_{2}\), on the interval [a, b]. Draw a line segment connecting the points \((x_{1}, f(x_{1}))\) and \((x_{2}, f(x_{2}))\). This produces a secant line. Kepler's Inequality states that the curvature of a convex function lies below the secant line. Hence, the line segment lies above the curve of the function.

Step 2 - Inferring Function Convexity from Second Derivative

According to the problem, we are given that \(f''(x) \geq 0\) for all \(x\) in the interval [a, b]. By the properties of derivatives, this implies that the function \(f(x)\) is convex on the interval [a, b].

Step 3 - Applying Formula to Find Position of Secant Line

The secant line formed by the points \((x_{1}, f(x_{1}))\) and \((x_{2}, f(x_{2}))\) has an equation given by \(f(x) = \frac{f(x_2) - f(x_1)}{x_2 - x_1}(x - x_1) + f(x_1)\). When we substitute \((x = \frac{x_1 + x_2}{2})\), we get \(f(\frac{x_1 + x_2}{2}) = \frac{f(x_1) + f(x_2)}{2}\).

Step 4 - Final result based on Function Convexity

Combining the information gathered from Steps 2 and 3, since \(f(x)\) is a convex function in the interval [a, b], the value of the function at any point within the interval [a, b] is less than or equal to the point on the secant line at the same \(x\)-value. Hence, \(f(\frac{x_1 + x_2}{2}) \leq \frac{f(x_1) + f(x_2)}{2}\) for \(x_1, x_2 \in [a, b]\), proving the given inequality.

Key Concepts

Kepler's Inequality

Kepler's Inequality is an interesting concept used to demonstrate properties of convex functions. To understand it, imagine a convex curve, which is a smooth line that doesn't bend inward. If you draw any secant line, that goes through any two points on this curve, the line will either touch or stay above the curve for all points between those two points.

This is essentially what Kepler's Inequality tells us: the average of the function's values at two points is always greater than or equal to the value of the function at the midpoint of those points. It's a wonderful way to show that convex functions tend to form a gentle bowl-shaped curve.

In practice, this means if you have two points on a curve and find the midpoint, the value of the function at this midpoint won't poke above the average value of the function at those two points. This basic yet powerful idea rests on the convexity of the function, making it a fundamental part of understanding how convex functions behave in relation to linearity.

Second Derivative Test

The second derivative test is a tool used to determine the nature of extrema (maximum or minimum points) of a function. In the context of Kepler's Inequality and convex functions, the second derivative test helps us establish the convexity of a function over a specific interval.

In mathematical terms, if a function's second derivative, denoted as \( f''(x) \), is greater than or equal to zero for all \( x \) in an interval, then the function \( f(x) \) is convex on that interval. A function is convex if its graph always lies below the tangent line at any point along the curve.

This property of convexity implies that a function doesn't suddenly dip below or spike above a line connecting any two points (the secant line) on the graph. Thus, in analyzing functions using the second derivative test, we derive insights into their broader geometric shapes beyond just the points of maxima or minima.

Secant Line

The secant line is a straight line connecting two points on the graph of a function. It's instrumental in various analytical tools, such as checking the convexity of a function. When dealing with convex functions, the secant line helps us compare the true curve of the function to a linear approximation between two points.

To visualize, imagine two points, \((x_1, f(x_1))\) and \((x_2, f(x_2))\), on the graph of a function. The line that goes through these two points is the secant line. For convex functions, this line sits above or touches the curve between these points, never dipping below it.

The equation of a secant line between two points \((x_1, f(x_1))\) and \((x_2, f(x_2))\) can be written as:

• \(f(x) = \frac{f(x_2) - f(x_1)}{x_2 - x_1}(x - x_1) + f(x_1)\)

This equation is especially useful because, by comparing the position of a secant line to the curve, we can assert important properties about the function, such as its convexity, through inequalities like the one proved in the problem.