You have a $20 coupon from the manufacturer that is good for the purchase of a cell phone. The store where you are purchasing your cell phone is offering a 20% discount on all cell phones. Let x represent the regular price of the cell phone. (a) Suppose only the 20% discount applies. Find a function f that models the purchase price of the cell phone as a function of the regular price x. f(x) = (b) Suppose only the $20 coupon applies. Find a function g that models the purchase price of the cell phone as a function of the sticker price x. g(x) = (c) If you can use the coupon and the discount, then the purchase price is either (f ∘ g)(x) or (g ∘ f)(x), depending on the order in which they are applied to the price. Find both (f ∘ g)(x) and (g ∘ f)(x). (f ∘ g)(x) = (g ∘ f)(x) = Which composition gives the lower price? Applying the coupon followed by the discount gives the lower price.Applying the discount followed by the coupon gives the lower price.
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.
You have a $20 coupon from the manufacturer that is good for the purchase of a cell phone. The store where you are purchasing your cell phone is offering a 20% discount on all cell phones. Let x represent the regular price of the cell phone.
(b) Suppose only the $20 coupon applies. Find a function g that models the purchase price of the cell phone as a function of the sticker price x.
(c) If you can use the coupon and the discount, then the purchase price is either
| (f ∘ g)(x) | = |
|
| (g ∘ f)(x) | = |
|
Which composition gives the lower price?
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