0.9 The profit in dollars from the sale of x expensive watches is P(x) = 0.04x- 2x+ 7x 5000. Find the marginal profit when (a) x= 100, (b) x= 2000, (c) x=5000, and (d) x= 10,000 (a) When x= 100, the marginal profit is S (Round to the nearest integer as needed.)
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.
![**Topic: Marginal Profit Calculation**
**The Problem:**
The profit in dollars from the sale of \( x \) expensive watches is given by the function:
\[ P(x) = 0.04x^2 - 2x + 7x^{0.9} - 5000 \]
**Objective:**
Find the marginal profit at different sales levels:
- (a) When \( x = 100 \)
- (b) When \( x = 2000 \)
- (c) When \( x = 5000 \)
- (d) When \( x = 10,000 \)
**Formula for Marginal Profit:**
Marginal profit is the derivative of the profit function \( P(x) \).
**Solution:**
For (a) When \( x = 100 \): Calculate the derivative, substitute \( x = 100 \) into the derivative, and evaluate. Round the result to the nearest integer.
**Note:** Complete the calculations for each specified \( x \) value as required to find the marginal profit in each case.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdd447044-2d2d-49a5-ae6a-64332b65b635%2Fd85c7469-fcf1-4832-a950-e70e8e16332b%2Fmr9nns_processed.jpeg&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
