Calculus Examples

$C = 6 x + \frac{500000}{x}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $6 x + \frac{500000}{x}$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [\frac{500000}{x}]$ .

$\frac{d}{d x} [6 x] + \frac{d}{d x} [\frac{500000}{x}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 1.1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 \frac{d}{d x} [x] + \frac{d}{d x} [\frac{500000}{x}]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$6 \cdot 1 + \frac{d}{d x} [\frac{500000}{x}]$

Step 1.1.2.3

Multiply $6$ by $1$ .

$6 + \frac{d}{d x} [\frac{500000}{x}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [\frac{500000}{x}]$ .

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Step 1.1.3.1

Since $500000$ is constant with respect to $x$ , the derivative of $\frac{500000}{x}$ with respect to $x$ is $500000 \frac{d}{d x} [\frac{1}{x}]$ .

$6 + 500000 \frac{d}{d x} [\frac{1}{x}]$

Step 1.1.3.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$6 + 500000 \frac{d}{d x} [x^{- 1}]$

Step 1.1.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 1$ .

$6 + 500000 (- x^{- 2})$

Step 1.1.3.4

Multiply $- 1$ by $500000$ .

$6 - 500000 x^{- 2}$

Step 1.1.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$6 - 500000 \frac{1}{x^{2}}$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Combine terms.

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Step 1.1.5.1.1

Combine $- 500000$ and $\frac{1}{x^{2}}$ .

$6 + \frac{- 500000}{x^{2}}$

Step 1.1.5.1.2

Move the negative in front of the fraction.

$6 - \frac{500000}{x^{2}}$

Step 1.1.5.2

Reorder terms.

$f' (x) = - \frac{500000}{x^{2}} + 6$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{500000}{x^{2}} + 6$ .

$- \frac{500000}{x^{2}} + 6$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{500000}{x^{2}} + 6 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{500000}{x^{2}} + 6 = 0$

Step 2.2

Subtract $6$ from both sides of the equation.

$- \frac{500000}{x^{2}} = - 6$

Step 2.3

Find the LCD of the terms in the equation.

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Step 2.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{2}, 1$

Step 2.3.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 2.4

Multiply each term in $- \frac{500000}{x^{2}} = - 6$ by $x^{2}$ to eliminate the fractions.

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Step 2.4.1

Multiply each term in $- \frac{500000}{x^{2}} = - 6$ by $x^{2}$ .

$- \frac{500000}{x^{2}} x^{2} = - 6 x^{2}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $x^{2}$ .

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Step 2.4.2.1.1

Move the leading negative in $- \frac{500000}{x^{2}}$ into the numerator.

$\frac{- 500000}{x^{2}} x^{2} = - 6 x^{2}$

Step 2.4.2.1.2

Cancel the common factor.

$\frac{- 500000}{x^{2}} x^{2} = - 6 x^{2}$

Step 2.4.2.1.3

Rewrite the expression.

$- 500000 = - 6 x^{2}$

Step 2.5

Solve the equation.

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Step 2.5.1

Rewrite the equation as $- 6 x^{2} = - 500000$ .

$- 6 x^{2} = - 500000$

Step 2.5.2

Divide each term in $- 6 x^{2} = - 500000$ by $- 6$ and simplify.

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Step 2.5.2.1

Divide each term in $- 6 x^{2} = - 500000$ by $- 6$ .

$\frac{- 6 x^{2}}{- 6} = \frac{- 500000}{- 6}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor of $- 6$ .

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Step 2.5.2.2.1.1

Cancel the common factor.

$\frac{- 6 x^{2}}{- 6} = \frac{- 500000}{- 6}$

Step 2.5.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 500000}{- 6}$

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Cancel the common factor of $- 500000$ and $- 6$ .

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Step 2.5.2.3.1.1

Factor $- 2$ out of $- 500000$ .

$x^{2} = \frac{- 2 (250000)}{- 6}$

Step 2.5.2.3.1.2

Cancel the common factors.

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Step 2.5.2.3.1.2.1

Factor $- 2$ out of $- 6$ .

$x^{2} = \frac{- 2 \cdot 250000}{- 2 \cdot 3}$

Step 2.5.2.3.1.2.2

Cancel the common factor.

$x^{2} = \frac{- 2 \cdot 250000}{- 2 \cdot 3}$

Step 2.5.2.3.1.2.3

Rewrite the expression.

$x^{2} = \frac{250000}{3}$

Step 2.5.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{250000}{3}}$

Step 2.5.4

Simplify $\pm \sqrt{\frac{250000}{3}}$ .

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Step 2.5.4.1

Rewrite $\sqrt{\frac{250000}{3}}$ as $\frac{\sqrt{250000}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{250000}}{\sqrt{3}}$

Step 2.5.4.2

Simplify the numerator.

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Step 2.5.4.2.1

Rewrite $250000$ as $500^{2}$ .

$x = \pm \frac{\sqrt{500^{2}}}{\sqrt{3}}$

Step 2.5.4.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \frac{500}{\sqrt{3}}$

Step 2.5.4.3

Multiply $\frac{500}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{500}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 2.5.4.4

Combine and simplify the denominator.

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Step 2.5.4.4.1

Multiply $\frac{500}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{500 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.5.4.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{500 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 2.5.4.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{500 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 2.5.4.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{500 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 2.5.4.4.5

Add $1$ and $1$ .

$x = \pm \frac{500 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.5.4.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.5.4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm \frac{500 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.5.4.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{500 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 2.5.4.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{500 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.5.4.4.6.4

Cancel the common factor of $2$ .

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Step 2.5.4.4.6.4.1

Cancel the common factor.

$x = \pm \frac{500 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.5.4.4.6.4.2

Rewrite the expression.

$x = \pm \frac{500 \sqrt{3}}{3^{1}}$

Step 2.5.4.4.6.5

Evaluate the exponent.

$x = \pm \frac{500 \sqrt{3}}{3}$

Step 2.5.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.5.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{500 \sqrt{3}}{3}$

Step 2.5.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{500 \sqrt{3}}{3}$

Step 2.5.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{500 \sqrt{3}}{3}, - \frac{500 \sqrt{3}}{3}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{500000}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 3.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 3.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

Evaluate $6 x + \frac{500000}{x}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{500 \sqrt{3}}{3}$ .

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Step 4.1.1

Substitute $\frac{500 \sqrt{3}}{3}$ for $x$ .

$6 (\frac{500 \sqrt{3}}{3}) + \frac{500000}{\frac{500 \sqrt{3}}{3}}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Cancel the common factor of $3$ .

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Step 4.1.2.1.1.1

Factor $3$ out of $6$ .

$3 (2) \frac{500 \sqrt{3}}{3} + \frac{500000}{\frac{500 \sqrt{3}}{3}}$

Step 4.1.2.1.1.2

Cancel the common factor.

$3 \cdot 2 \frac{500 \sqrt{3}}{3} + \frac{500000}{\frac{500 \sqrt{3}}{3}}$

Step 4.1.2.1.1.3

Rewrite the expression.

$2 (500 \sqrt{3}) + \frac{500000}{\frac{500 \sqrt{3}}{3}}$

Step 4.1.2.1.2

Multiply $500$ by $2$ .

$1000 \sqrt{3} + \frac{500000}{\frac{500 \sqrt{3}}{3}}$

Step 4.1.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$1000 \sqrt{3} + 500000 \frac{3}{500 \sqrt{3}}$

Step 4.1.2.1.4

Cancel the common factor of $500$ .

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Step 4.1.2.1.4.1

Factor $500$ out of $500000$ .

$1000 \sqrt{3} + 500 (1000) \frac{3}{500 \sqrt{3}}$

Step 4.1.2.1.4.2

Factor $500$ out of $500 \sqrt{3}$ .

$1000 \sqrt{3} + 500 (1000) \frac{3}{500 (\sqrt{3})}$

Step 4.1.2.1.4.3

Cancel the common factor.

$1000 \sqrt{3} + 500 \cdot 1000 \frac{3}{500 \sqrt{3}}$

Step 4.1.2.1.4.4

Rewrite the expression.

$1000 \sqrt{3} + 1000 \frac{3}{\sqrt{3}}$

Step 4.1.2.1.5

Combine $1000$ and $\frac{3}{\sqrt{3}}$ .

$1000 \sqrt{3} + \frac{1000 \cdot 3}{\sqrt{3}}$

Step 4.1.2.1.6

Multiply $1000$ by $3$ .

$1000 \sqrt{3} + \frac{3000}{\sqrt{3}}$

Step 4.1.2.1.7

Multiply $\frac{3000}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$1000 \sqrt{3} + \frac{3000}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 4.1.2.1.8

Combine and simplify the denominator.

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Step 4.1.2.1.8.1

Multiply $\frac{3000}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$1000 \sqrt{3} + \frac{3000 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.1.2.1.8.2

Raise $\sqrt{3}$ to the power of $1$ .

$1000 \sqrt{3} + \frac{3000 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 4.1.2.1.8.3

Raise $\sqrt{3}$ to the power of $1$ .

$1000 \sqrt{3} + \frac{3000 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 4.1.2.1.8.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1000 \sqrt{3} + \frac{3000 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.1.2.1.8.5

Add $1$ and $1$ .

$1000 \sqrt{3} + \frac{3000 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 4.1.2.1.8.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.1.2.1.8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$1000 \sqrt{3} + \frac{3000 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 4.1.2.1.8.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$1000 \sqrt{3} + \frac{3000 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 4.1.2.1.8.6.3

Combine $\frac{1}{2}$ and $2$ .

$1000 \sqrt{3} + \frac{3000 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.1.2.1.8.6.4

Cancel the common factor of $2$ .

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Step 4.1.2.1.8.6.4.1

Cancel the common factor.

$1000 \sqrt{3} + \frac{3000 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.1.2.1.8.6.4.2

Rewrite the expression.

$1000 \sqrt{3} + \frac{3000 \sqrt{3}}{3^{1}}$

Step 4.1.2.1.8.6.5

Evaluate the exponent.

$1000 \sqrt{3} + \frac{3000 \sqrt{3}}{3}$

Step 4.1.2.1.9

Cancel the common factor of $3000$ and $3$ .

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Step 4.1.2.1.9.1

Factor $3$ out of $3000 \sqrt{3}$ .

$1000 \sqrt{3} + \frac{3 (1000 \sqrt{3})}{3}$

Step 4.1.2.1.9.2

Cancel the common factors.

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Step 4.1.2.1.9.2.1

Factor $3$ out of $3$ .

$1000 \sqrt{3} + \frac{3 (1000 \sqrt{3})}{3 (1)}$

Step 4.1.2.1.9.2.2

Cancel the common factor.

$1000 \sqrt{3} + \frac{3 (1000 \sqrt{3})}{3 \cdot 1}$

Step 4.1.2.1.9.2.3

Rewrite the expression.

$1000 \sqrt{3} + \frac{1000 \sqrt{3}}{1}$

Step 4.1.2.1.9.2.4

Divide $1000 \sqrt{3}$ by $1$ .

$1000 \sqrt{3} + 1000 \sqrt{3}$

Step 4.1.2.2

Add $1000 \sqrt{3}$ and $1000 \sqrt{3}$ .

$2000 \sqrt{3}$

Step 4.2

Evaluate at $x = - \frac{500 \sqrt{3}}{3}$ .

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Step 4.2.1

Substitute $- \frac{500 \sqrt{3}}{3}$ for $x$ .

$6 (- \frac{500 \sqrt{3}}{3}) + \frac{500000}{- \frac{500 \sqrt{3}}{3}}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1.1

Move the leading negative in $- \frac{500 \sqrt{3}}{3}$ into the numerator.

$6 \frac{- 500 \sqrt{3}}{3} + \frac{500000}{- \frac{500 \sqrt{3}}{3}}$

Step 4.2.2.1.1.2

Factor $3$ out of $6$ .

$3 (2) \frac{- 500 \sqrt{3}}{3} + \frac{500000}{- \frac{500 \sqrt{3}}{3}}$

Step 4.2.2.1.1.3

Cancel the common factor.

$3 \cdot 2 \frac{- 500 \sqrt{3}}{3} + \frac{500000}{- \frac{500 \sqrt{3}}{3}}$

Step 4.2.2.1.1.4

Rewrite the expression.

$2 (- 500 \sqrt{3}) + \frac{500000}{- \frac{500 \sqrt{3}}{3}}$

Step 4.2.2.1.2

Multiply $- 500$ by $2$ .

$- 1000 \sqrt{3} + \frac{500000}{- \frac{500 \sqrt{3}}{3}}$

Step 4.2.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$- 1000 \sqrt{3} + 500000 (- \frac{3}{500 \sqrt{3}})$

Step 4.2.2.1.4

Cancel the common factor of $500$ .

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Step 4.2.2.1.4.1

Move the leading negative in $- \frac{3}{500 \sqrt{3}}$ into the numerator.

$- 1000 \sqrt{3} + 500000 \frac{- 3}{500 \sqrt{3}}$

Step 4.2.2.1.4.2

Factor $500$ out of $500000$ .

$- 1000 \sqrt{3} + 500 (1000) \frac{- 3}{500 \sqrt{3}}$

Step 4.2.2.1.4.3

Factor $500$ out of $500 \sqrt{3}$ .

$- 1000 \sqrt{3} + 500 (1000) \frac{- 3}{500 (\sqrt{3})}$

Step 4.2.2.1.4.4

Cancel the common factor.

$- 1000 \sqrt{3} + 500 \cdot 1000 \frac{- 3}{500 \sqrt{3}}$

Step 4.2.2.1.4.5

Rewrite the expression.

$- 1000 \sqrt{3} + 1000 \frac{- 3}{\sqrt{3}}$

Step 4.2.2.1.5

Combine $1000$ and $\frac{- 3}{\sqrt{3}}$ .

$- 1000 \sqrt{3} + \frac{1000 \cdot - 3}{\sqrt{3}}$

Step 4.2.2.1.6

Multiply $1000$ by $- 3$ .

$- 1000 \sqrt{3} + \frac{- 3000}{\sqrt{3}}$

Step 4.2.2.1.7

Move the negative in front of the fraction.

$- 1000 \sqrt{3} - \frac{3000}{\sqrt{3}}$

Step 4.2.2.1.8

Multiply $\frac{3000}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$- 1000 \sqrt{3} - (\frac{3000}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 4.2.2.1.9

Combine and simplify the denominator.

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Step 4.2.2.1.9.1

Multiply $\frac{3000}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$- 1000 \sqrt{3} - \frac{3000 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.2.2.1.9.2

Raise $\sqrt{3}$ to the power of $1$ .

$- 1000 \sqrt{3} - \frac{3000 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 4.2.2.1.9.3

Raise $\sqrt{3}$ to the power of $1$ .

$- 1000 \sqrt{3} - \frac{3000 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 4.2.2.1.9.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 1000 \sqrt{3} - \frac{3000 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.2.2.1.9.5

Add $1$ and $1$ .

$- 1000 \sqrt{3} - \frac{3000 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 4.2.2.1.9.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.2.2.1.9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$- 1000 \sqrt{3} - \frac{3000 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 4.2.2.1.9.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 1000 \sqrt{3} - \frac{3000 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 4.2.2.1.9.6.3

Combine $\frac{1}{2}$ and $2$ .

$- 1000 \sqrt{3} - \frac{3000 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.2.2.1.9.6.4

Cancel the common factor of $2$ .

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Step 4.2.2.1.9.6.4.1

Cancel the common factor.

$- 1000 \sqrt{3} - \frac{3000 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.2.2.1.9.6.4.2

Rewrite the expression.

$- 1000 \sqrt{3} - \frac{3000 \sqrt{3}}{3^{1}}$

Step 4.2.2.1.9.6.5

Evaluate the exponent.

$- 1000 \sqrt{3} - \frac{3000 \sqrt{3}}{3}$

Step 4.2.2.1.10

Cancel the common factor of $3000$ and $3$ .

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Step 4.2.2.1.10.1

Factor $3$ out of $3000 \sqrt{3}$ .

$- 1000 \sqrt{3} - \frac{3 (1000 \sqrt{3})}{3}$

Step 4.2.2.1.10.2

Cancel the common factors.

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Step 4.2.2.1.10.2.1

Factor $3$ out of $3$ .

$- 1000 \sqrt{3} - \frac{3 (1000 \sqrt{3})}{3 (1)}$

Step 4.2.2.1.10.2.2

Cancel the common factor.

$- 1000 \sqrt{3} - \frac{3 (1000 \sqrt{3})}{3 \cdot 1}$

Step 4.2.2.1.10.2.3

Rewrite the expression.

$- 1000 \sqrt{3} - \frac{1000 \sqrt{3}}{1}$

Step 4.2.2.1.10.2.4

Divide $1000 \sqrt{3}$ by $1$ .

$- 1000 \sqrt{3} - (1000 \sqrt{3})$

Step 4.2.2.1.11

Multiply $1000$ by $- 1$ .

$- 1000 \sqrt{3} - 1000 \sqrt{3}$

Step 4.2.2.2

Subtract $1000 \sqrt{3}$ from $- 1000 \sqrt{3}$ .

$- 2000 \sqrt{3}$

Step 4.3

Evaluate at $x = 0$ .

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Step 4.3.1

Substitute $0$ for $x$ .

$6 (0) + \frac{500000}{0}$

Step 4.3.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.4

List all of the points.

$(\frac{500 \sqrt{3}}{3}, 2000 \sqrt{3}), (- \frac{500 \sqrt{3}}{3}, - 2000 \sqrt{3})$

Step 5