2. Let f be a continuous function defined on a closed, bounded interval I = [a, b]. Assume that f is one-to-one. Let m (M, respectively) be the minimum (maximum, respectively) of f. Then by problem 1, we know that either f(a) = m and f(b) = M; or f(a) = M and f(b) = m. If f (a) = m and f(b) = M, then show that f is strictly monotone increasing. If f(a) monotone decreasing. %3D M and f(a) = m, then show that f is strictly %3D Hint: you need to use the Intermediate Value Theorem and argue by contradition. Remark: Problem 2 basically says the only way for a continuous func- tion to be one-to-one is to be strictly monotone. In other words, only strictly monotone continuous functions can have inverse. This is actually a Theorem in our text.
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![2. Let f be a continuous function defined on a closed, bounded interval I = [a, 6].
Assume that f is one-to-one. Let m (M, respectively) be the minimum (maximum,
respectively) of f. Then by problem 1, we know that either f(a) = m and f(b) = M;
or f(a) = M and f(b) = m. If f(a)
monotone increasing. If f(a) = M and f(a)
monotone decreasing.
%3D
%3D
= m and f(b) = M, then show that f is strictly
= m, then show that f is strictly
%3D
Hint: you need to use the Intermediate Value Theorem and argue by contradition.
Remark: Problem 2 basically says the only way for a continuous func-
tion to be one-to-one is to be strictly monotone. In other words, only
strictly monotone continuous functions can have inverse. This is actually
a Theorem in our text.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe6dba997-101e-43e2-95cb-d4dc7d29171d%2F7c0a2570-7b83-40a3-bbaf-294f3180a7a3%2Fpwcpchn_processed.jpeg&w=3840&q=75)
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