(a) Consider the function f(x) = 1/x – 2 and the interval [a, b] = [-1, ]. Is it true that f(a)f(b) < 0? Is [a, b] an interval of uncertainty for a zero of f(x)? Explain your answer. (b) Let f(x) be a continuous nonlinear function. Let Ik = [ak, br] be an interval such that f(ax) < 0 and f(bx) > 0. Compute an linear model function m(x) that has the value sign(f(x)) at ark and bg. Show how the zero xL of mä(x) can be used to reduce the length of the interval of uncertainty for a zero of f.
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.
![(a) Consider the function f(x) = 1/x – 2 and the interval [a, b] = [–1, ]. Is it true that
f(a)f(b) < 0? Is [a, b] an interval of uncertainty for a zero of f(x)? Explain your
answer.
(b) Let f(x) be a continuous nonlinear function. Let Ik = [ak, br] be an interval such that
f(ax) < 0 and f(br) > 0. Compute an linear model function mr(x) that has the value
sign(f(x)) at ark and bg. Show how the zero xL of mä(x) can be used to reduce the
length of the interval of uncertainty for a zero of f.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F833481a2-df8c-4805-95a2-f24b64ba619f%2Ffaa5d778-2336-4d47-ad43-ebe72ec1072a%2Fdiovxnl_processed.png&w=3840&q=75)
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