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(x) = 4x²(x – 2) + 6.
Find the critical points c that correspond to local minima.
(Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. Enter DNE if
there are no critical points.)
c =
Find the critical points c that correspond to local maxima.
(Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. Enter DNE if
there are no critical points.)
c =
Find values at which the points of inflection occur.
(Use symbolic notation and fractions where needed. Give your answer as a comma separated list. Enter DNE if there are no
points of inflection.)
Determine the interval on which f is concave up.
(Use symbolic notation and fractions where needed. Give your answer as interval in the form (*, *). Use the symbol o for
infinity, U for combining intervals, and an appropriate type of parenthesis "(", ")", "[", "]" depending on whether the interval is
open or closed. Enter Ø if the interval is empty.)
x E
Determine the interval on which f is concave down.
(Use symbolic notation and fractions where needed. Give your answer as interval in the form (*, *). Use the symbol o for
infinity, U for combining intervals, and an appropriate type of parenthesis "(", ")", "[", "]" depending on whether the interval is
open or closed. Enter Ø if the interval is empty.)
x E](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd31682c5-2fc9-464e-9dc7-0f0389d48dd3%2Fd396a539-a2f4-41b1-9551-1ac22cc59c80%2F3glm4e_processed.png&w=3840&q=75)
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