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.
![Let \( f \) be the function defined as follows:
\[
f(x) =
\begin{cases}
2x + 1, & \text{for } x \leq 2 \\
\frac{1}{2}x^2 + k, & \text{for } x > 2
\end{cases}
\]
a) For what values of \( k \) will \( f \) be continuous at \( x = 2 \)? Justify your answer.
b) Using the value of \( k \) found in part a), determine whether \( f \) is differentiable at \( x = 2 \). Use the definition of the derivative to justify your answer.
c) Let \( k = 4 \). Determine whether \( f \) is differentiable at \( x = 2 \). Justify your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F83b86ac8-1214-407a-8ec2-632d5098b00d%2Fecb07a65-7227-425f-a41e-31aaeac69c1c%2F41xu67g.jpeg&w=3840&q=75)
Given that,
(a)
If the function is continuous at x = 2, one-sided limits are equal.
That is,
The function is continuous at x = 2 for k = 3.
(b)
A function is differentiable at x= c if and only if the left and right derivatives at c both exist and are equal.
Substitute k = 3 in the given function.
To find left-sided derivative at x = 2, find derivative of 2x + 1 at x = 2.
Definition of derivative of f(x) at point x = a is
The left-sided derivative at x = 2 is 2.
To find left-sided derivative at x = 2, find derivative of at x = 2.
The right-sided derivative at x = 2 is 2.
Here the left and right derivatives at x= 2 both exist and are equal.
The function is differentiable at x = 2.
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