The problem consists of four parts: Q-3 a) Given the demand function $P = 500 - 0.2x$ and the cost function $C = 25x + 10000$, we need to determine the output $x$ at which the profit is maximized and the price $P$ at that output. Q-3 b) Show that the function $y = x^3 - 3x^2$ has a point of inflection at the point (1, -2). Q-4 We are given the demand function $D(x) = 25 - 5x + \frac{x^2}{4}$ and supply function $S(x) = 5x + \frac{x^2}{4}$. We need to estimate: i) The market price the item would be sold ii) The Consumers' Surplus iii) The Producers' Surplus. Q-5 a) Simplify the following integrals: i) $\int (x^3 + 2)^2 \cdot 3x^2 dx$ ii) $\int \frac{x^2 - 3x + \sqrt[3]{x} + 7}{\sqrt{x}} dx$

1. Problem Description

The problem consists of four parts:

Q-3 a) Given the demand function $P = 500 - 0.2x$ and the cost function $C = 25x + 10000$, we need to determine the output $x$ at which the profit is maximized and the price $P$ at that output.

Q-3 b) Show that the function $y = x^3 - 3x^2$ has a point of inflection at the point (1, -2).

Q-4 We are given the demand function $D(x) = 25 - 5x + \frac{x^2}{4}$ and supply function $S(x) = 5x + \frac{x^2}{4}$. We need to estimate:

i) The market price the item would be sold

ii) The Consumers' Surplus

iii) The Producers' Surplus.

Q-5 a) Simplify the following integrals:

i) $\int (x^3 + 2)^2 \cdot 3x^2 dx$

ii) $\int \frac{x^2 - 3x + \sqrt[3]{x} + 7}{\sqrt{x}} dx$

2. Solution Steps

Q-3 a)

First, find the revenue function $R(x)$. Revenue is price times quantity, so $R(x) = Px = (500 - 0.2x)x = 500x - 0.2x^2$.

The profit function $\pi(x)$ is revenue minus cost: $\pi(x) = R(x) - C(x) = (500x - 0.2x^2) - (25x + 10000) = 475x - 0.2x^2 - 10000$.

To maximize profit, we take the derivative of the profit function with respect to $x$ and set it equal to zero:

$\frac{d\pi}{dx} = 475 - 0.4x = 0$.

Solving for $x$, we get $0.4x = 475$, so $x = \frac{475}{0.4} = 1187.5$.

To find the price, plug $x = 1187.5$ into the demand function: $P = 500 - 0.2(1187.5) = 500 - 237.5 = 262.5$.

Q-3 b)

To find the point of inflection, we need to find the second derivative of $y = x^3 - 3x^2$ and set it equal to zero.

First derivative: $y' = 3x^2 - 6x$.

Second derivative: $y'' = 6x - 6$.

Set $y'' = 0$: $6x - 6 = 0$, which means $6x = 6$, so $x = 1$.

Now, let's find the $y$-value when $x = 1$: $y = (1)^3 - 3(1)^2 = 1 - 3 = -2$.

Therefore, the point of inflection is indeed (1, -2).

Q-4

i) To find the market price, we set the demand and supply functions equal to each other:

$D(x) = S(x)$

$25 - 5x + \frac{x^2}{4} = 5x + \frac{x^2}{4}$

$25 - 5x = 5x$

$25 = 10x$

$x = \frac{25}{10} = 2.5$

Now, we plug $x = 2.5$ into either the demand or supply function. Let's use the supply function:

$S(2.5) = 5(2.5) + \frac{(2.5)^2}{4} = 12.5 + \frac{6.25}{4} = 12.5 + 1.5625 = 14.0625$.

So, the market price is $14.

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ii) Consumer Surplus (CS) is given by $\int_0^{x_e} D(x) dx - P_e x_e$, where $x_e$ is the equilibrium quantity and $P_e$ is the equilibrium price.

$\int_0^{2.5} (25 - 5x + \frac{x^2}{4}) dx = [25x - \frac{5}{2}x^2 + \frac{x^3}{12}]_0^{2.5} = 25(2.5) - \frac{5}{2}(2.5)^2 + \frac{(2.5)^3}{12} = 62.5 - 15.625 + 1.302 = 48.177$

$CS = 48.177 - (14.0625)(2.5) = 48.177 - 35.15625 = 13.02075$.

iii) Producer Surplus (PS) is given by $P_e x_e - \int_0^{x_e} S(x) dx$.

$\int_0^{2.5} (5x + \frac{x^2}{4}) dx = [\frac{5}{2}x^2 + \frac{x^3}{12}]_0^{2.5} = \frac{5}{2}(2.5)^2 + \frac{(2.5)^3}{12} = 15.625 + 1.302 = 16.927$.

$PS = 35.15625 - 16.927 = 18.22925$.

Q-5 a)

i) Let $u = x^3 + 2$. Then $\frac{du}{dx} = 3x^2$, so $du = 3x^2 dx$. Thus,

$\int (x^3 + 2)^2 \cdot 3x^2 dx = \int u^2 du = \frac{1}{3}u^3 + C = \frac{1}{3}(x^3 + 2)^3 + C$.

ii) $\int \frac{x^2 - 3x + \sqrt[3]{x} + 7}{\sqrt{x}} dx = \int \frac{x^2}{x^{1/2}} - \frac{3x}{x^{1/2}} + \frac{x^{1/3}}{x^{1/2}} + \frac{7}{x^{1/2}} dx = \int x^{3/2} - 3x^{1/2} + x^{-1/6} + 7x^{-1/2} dx$.

Using the power rule for integration, we have:

$= \frac{x^{5/2}}{5/2} - 3\frac{x^{3/2}}{3/2} + \frac{x^{5/6}}{5/6} + 7\frac{x^{1/2}}{1/2} + C = \frac{2}{5}x^{5/2} - 2x^{3/2} + \frac{6}{5}x^{5/6} + 14x^{1/2} + C$.

3. Final Answer

Q-3 a) The output at which the profit is maximized is $x = 1187.5$, and the price is $P = 262.5$.

Q-3 b) The function $y = x^3 - 3x^2$ has a point of inflection at (1, -2).

Q-4 i) The market price is $14.0625$.

Q-4 ii) The Consumer Surplus is $13.02075$.

Q-4 iii) The Producer Surplus is $18.22925$.

Q-5 a)

i) $\int (x^3 + 2)^2 \cdot 3x^2 dx = \frac{1}{3}(x^3 + 2)^3 + C$.

ii) $\int \frac{x^2 - 3x + \sqrt[3]{x} + 7}{\sqrt{x}} dx = \frac{2}{5}x^{5/2} - 2x^{3/2} + \frac{6}{5}x^{5/6} + 14x^{1/2} + C$.