Question

involving the function and its derivatives. Here we look at some applications of the theorem for functions of one and two variables. a. Any continuous and differentiable function of a single variable, \(f(x),\) can be approximated near the point \(a\) by the formula \\[ f(x)=f(a)+f^{\prime}(a)(x-a)+0.5 f^{\prime \prime}(a)(x-a)^{2}+\text { terms in } f^{\prime \prime \prime}, f^{\prime \prime \prime \prime}, \ldots \\] Using only the first three of these terms results in a quadratic Taylor approximation. Use this approximation together with the definition of concavity given in Equation 2.85 to show that any concave function must lie on or below the tangent to the function at point \(a\) b. The quadratic Taylor approximation for any function of two variables, \(f(x, y),\) near the point \((a, b)\) is given by \\[ \begin{aligned} f(x, y)=& f(a, b)+f_{1}(a, b)(x-a)+f_{2}(a, b)(y-b) \\ &+0.5\left[f_{11}(a, b)(x-a)^{2}+2 f_{12}(a, b)(x-a)(y-b)+f_{22}(y-b)^{2}\right] \end{aligned} \\] Use this approximation to show that any concave function (as defined by Equation 2.98 ) must lie on or below its tangent plane at \((a, b)\).

Short answer

Question: Prove that any concave function of one and two variables must lie on or below its tangent line (for one variable) or tangent plane (for two variables) at a given point using the quadratic Taylor approximation. Answer: By using the quadratic Taylor approximation and ignoring higher-order terms, we can compare the function value f(x) or f(x,y) to its respective tangent line g(x) or tangent plane h(x,y) equation. For concave functions in one variable, the second derivative must be less than or equal to zero, proving that f(x) must lie on or below g(x). In the case of concave functions of two variables, the Hessian matrix must be negative semi-definite, which implies that f(x, y) must lie on or below h(x, y).

Step-by-step solution

Write down the quadratic Taylor approximation for a single variable function.

Given the quadratic Taylor approximation for a single variable function f(x) near point a: \\[ f(x) = f(a) + f'(a)(x-a) + 0.5f''(a)(x-a)^2 + \text{ terms in }f'''(a), f''''(a), \ldots \\]

Ignore the higher order terms for simplicity.

We only need the first three terms of the Taylor approximation for this proof, so we can simplify it to: \\[ f(x) \approx f(a) + f'(a)(x-a) + 0.5f''(a)(x-a)^2 \\]

Define the tangent line at point a.

The tangent line at point a is given by the equation: \\[ g(x) = f(a) + f'(a)(x-a) \\]

Show that the function lies on or below the tangent line.

Since the function is concave, its second derivative must be less than or equal to zero (from the definition of concavity): \\[ f''(a) \leq 0 \\] Now, let's compare f(x) and g(x) from Step 2 and Step 3: \\[ f(x) \approx f(a) + f'(a)(x-a) + 0.5f''(a)(x-a)^2 \leq f(a) + f'(a)(x-a) = g(x) \\] Hence, any concave function must lie on or below the tangent to the function at point a. b. Proving concave functions of two variables lie on or below their tangent plane:

Write down the quadratic Taylor approximation for a two-variable function.

Given the quadratic Taylor approximation for a two-variable function f(x,y) near point (a, b): \\[ \begin{aligned} f(x, y) \approx & f(a, b) + f_{1}(a, b)(x-a) + f_{2}(a, b)(y-b) \\ & +0.5\left[f_{11}(a, b)(x-a)^{2} + 2 f_{12}(a, b)(x-a)(y-b) + f_{22}(a, b)(y-b)^{2}\right] \end{aligned} \\]

Define the tangent plane at point (a, b).

The tangent plane at point (a, b) is given by the equation: \\[ h(x, y) = f(a, b) + f_{1}(a, b)(x-a) + f_{2}(a, b)(y-b) \\]

Show that the function lies on or below the tangent plane.

A concave function of two variables has a negative semi-definite Hessian (from the definition of concavity): \\[ H(f) = \begin{bmatrix} f_{11}(a,b) & f_{12}(a,b) \\ f_{12}(a,b) & f_{22}(a,b) \end{bmatrix} \preceq 0 \\] Using the quadratic Taylor approximation and the Hessian matrix condition for concavity: \\[ \begin{aligned} f(x, y) \approx & f(a, b) + f_{1}(a, b)(x-a) + f_{2}(a, b)(y-b) \\ & +0.5\left[f_{11}(a, b)(x-a)^{2} + 2 f_{12}(a, b)(x-a)(y-b) + f_{22}(a, b)(y-b)^{2}\right] \\ & \leq f(a, b) + f_{1}(a, b)(x-a) + f_{2}(a, b)(y-b) \\ & = h(x, y) \end{aligned} \\] Hence, any concave function of two variables must lie on or below its tangent plane at point (a, b).

Key concepts

Concavity

In calculus, concavity refers to the curvature of a graph of a function. If a function is concave down, it resembles an upside-down bowl. Conversely, a concave up function looks like a right-side up bowl. The mathematical center of understanding concavity lies in the second derivative. - A function is concave down if its second derivative is negative, indicating that the slope of the tangent line is decreasing. - A function is concave up if its second derivative is positive, suggesting an increasing slope. When analyzing functions, recognizing whether they are concave up or down helps us determine their nature and stability in applications. For example, in economics, a concave down function might represent a diminishing return on investment, while a concave up function might indicate accelerating growth.

Tangent Plane

A tangent plane is the generalization of the tangent line concept to multivariable calculus. Just as the tangent line can touch a curve at a single point, the tangent plane does the same for surfaces in a three-dimensional space.- The equation of the tangent plane at a point \(a, b\) for a function \(f(x, y)\) is derived from the gradient, which provides the direction of steepest ascent.- The formula is: \ h(x, y) = f(a, b) + f_{1}(a, b)(x-a) + f_{2}(a, b)(y-b) \.This means at the point \(a, b\), the tangent plane approximates the surface locally. In practical terms, this approximation allows for easier calculations when examining behavior near a particular point on a surface.

Hessian Matrix

The Hessian matrix is essential in studying functions of multiple variables. It consists of second-order partial derivatives and helps determine the local curvature of a function. It plays a crucial role in optimization problems, especially with functions of more than one variable.- For a function \(f(x, y)\), the Hessian matrix is given by \ H(f) = \begin{bmatrix} f_{11}(a,b) & f_{12}(a,b) \ f_{12}(a,b) & f_{22}(a,b) \end{bmatrix} \.- This matrix helps in determining whether a function is convex, concave, or saddle at a given point.If the Hessian matrix is negative semi-definite, the function is concave at the point \(a, b\). This information is particularly useful in confirming whether a given surface behaves the way we expect, such as whether it is conveying a maximum or minimum point on certain axes.