(a) what's the derivative of f(x)= 1/((x-1)(x-3))2 (b) Find all (three) locations where the derivative of f is zero or does not exist. (c) Identify the location of the (only) critical point of f. You can look this up on Wolfram Alpha to check your answer. Why aren’t all of the points in part (b) critical points? (d) Suppose now you make a sign chart with just the one and only critical point (which is not the right thing to do). Check the sign of f 0 (x) at the values x = 0 and x = 4. According to this sign chart, what would you conclude about the classification of the critical point as a max or min? (e) Now draw the correct sign chart with all three points where f'(x) = 0 or DNE, and determine whether the critical point corresponds to a max or min. (f) Explain in general why it is not sufficient to draw a sign chart using only the critical points. Based on part (e), what should you do instead? (g) Use some technology to take a look at the plot of f(x). Why don’t the points (1, f(1)) and (3, f(3)) correspond to local maxima? Can you use the sign chart to classify these points?
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) what's the derivative of
f(x)= 1/((x-1)(x-3))2
(b) Find all (three) locations where the derivative of f is zero or does not exist.
(c) Identify the location of the (only) critical point of f. You can look this up on Wolfram Alpha to check your answer. Why aren’t all of the points in part (b) critical points?
(d) Suppose now you make a sign chart with just the one and only critical point (which is not the right thing to do). Check the sign of f 0 (x) at the values x = 0 and x = 4. According to this sign chart, what would you conclude about the classification of the critical point as a max or min?
(e) Now draw the correct sign chart with all three points where f'(x) = 0 or DNE, and determine whether the critical point corresponds to a max or min.
(f) Explain in general why it is not sufficient to draw a sign chart using only the critical points. Based on part (e), what should you do instead?
(g) Use some technology to take a look at the plot of f(x). Why don’t the points (1, f(1)) and (3, f(3)) correspond to
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