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8. Optimization of functions of one variable

Faculty of Economics and Business
University of Zagreb

8.1Introduction

In this chapter, we are going to see how to use the derivative in order to answer the following questions:

The answers to all of these questions are going to be algorithmic in nature, meaning we will state the algorithm which will lead us to answers to the questions above. The main thing that we’ll use to derive those algorithms is the relationship between the function ff and its derivatives.

The relationship between the function ff and its derivatives.

8.2Local extrema

Graph of the function f(x) = e^x -x. The point x = 0 is a local minimum of the function f.

Graph of the function f(x)=ex−x.f(x) = e^x -x. The point x=0x = 0 is a local minimum of the function f.f.

8.3Intervals of monotonicity

Graph of the function \displaystyle f(x) = \frac{x^2}{4-x}.

Graph of the function f(x)=x24−x.\displaystyle f(x) = \frac{x^2}{4-x}.

Common mistake on the Midterm/Exam

A common mistake that students make when they are asked to define an increasing function is that they say that a function is increasing if f′(x)>0.f'(x) > 0. That statement is not correct!
By definition, a function is increasing if

x<y  ⟹  f(x)<f(y)x < y \implies f(x) < f(y)

Of course, if a function ff is increasing, then f′(x)>0,f'(x) > 0, but having a positive derivative is not the definition of an increasing function, but rather a consequence of the definition.
Keep in mind what is defining property of something, and what is a consequence of the definition!

8.4Convexity and concavity

Graph of the function \displaystyle f(x) = x^3-4x.

Graph of the function f(x)=x3−4x.\displaystyle f(x) = x^3-4x.

Part 2. Functions of one variable
7. Differential of a function
Part 2. Functions of one variable
9. Elasticity and economic quantities