Given that the total profit in AED by selling x items is given by  ﻿P left parenthesis x right parenthesis space equals space fraction numerator 2 x cubed plus 8 x minus 10 over denominator x end fraction﻿ 
Find the marginal profit at x = 5

The marginal profit is found by differentiating the profit function \( P(x) \) with respect to \( x \), and then evaluating the derivative at \( x = 5 \). The given profit function is:

\[
P(x) = \frac{2x^3 + 8x - 10}{x}
\]

To find the marginal profit, we need to compute \( P'(x) \).

### Step 1: Simplify the profit function
First, simplify the given function by dividing each term in the numerator by \( x \):

\[
P(x) = 2x^2 + 8 - \frac{10}{x}
\]

### Step 2: Differentiate \( P(x) \)
Now, differentiate \( P(x) \) with respect to \( x \):

\[
P'(x) = \frac{d}{dx} \left( 2x^2 \right) + \frac{d}{dx} \left( 8 \right) + \frac{d}{dx} \left( -\frac{10}{x} \right)
\]

This results in:

\[
P'(x) = 4x + 0 + \frac{10}{x^2}
\]

So, the derivative is:

\[
P'(x) = 4x + \frac{10}{x^2}
\]

### Step 3: Evaluate at \( x = 5 \)

Substitute \( x = 5 \) into the derivative:

\[
P'(5) = 4(5) + \frac{10}{5^2} = 20 + \frac{10}{25} = 20 + 0.4 = 20.4
\]

Thus, the marginal profit at \( x = 5 \) is **20.4 AED**.

Would you like more details or further explanations?

Here are 5 related questions for further practice:
1. What is the profit at \( x = 

Given that the total profit in AED by selling x items is given by P(x) = (2x^3 + 8x - 10)/x, find the marginal profit at x = 5.

This problem involves finding the marginal profit by differentiating the profit function P(x) = (2x^3 + 8x - 10)/x and evaluating it at x = 5. Learn step-by-step how to simplify the function, apply the derivative, and calculate the marginal profit. Suitable for Grades 11-12 students studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation