Question

How do you find the value of $ x $ that gives the minimum average cost, if the cost of producing $ x $ units of a certain product is given by $ C=10,000+5x+\left( \dfrac{1}{9} \right){{x}^{2}} $ ?

Answer 1

Hint: In this problem we need to calculate the minimum value of $ x $ where the function is given by $ C=10,000+5x+\left( \dfrac{1}{9} \right){{x}^{2}} $ . We know that the function $ f\left( x \right) $ has a minimum value at $ x $ where $ {{f}^{'}}\left( x \right)=0 $ . So, we will calculate the derivative of the derivative of the given function and equate it to zero. Now we will simplify the obtained equation and calculate the value of $ x $ .

Complete step by step answer:
Given equation $ C=10,000+5x+\left( \dfrac{1}{9} \right){{x}^{2}} $
Differentiating the above equation with respect to $ x $ , then we will get
 $ {{C}^{'}}={{\left( 10,000+5x+\dfrac{1}{9}{{x}^{2}} \right)}^{'}} $
Distributing the differentiation over each term, then we will get
 $ {{C}^{'}}={{\left( 10,000 \right)}^{'}}+{{\left( 5x \right)}^{'}}+{{\left( \dfrac{1}{9}{{x}^{2}} \right)}^{'}} $
We know that the differentiation of constant with respect to $ x $ gives zero, then we will get
 $ {{C}^{'}}=0+{{\left( 5x \right)}^{'}}+{{\left( \dfrac{1}{9}{{x}^{2}} \right)}^{'}} $
We have the differentiation rule $ \dfrac{d}{dx}\left[ af\left( x \right) \right]=a\dfrac{d}{dx}\left[ f\left( x \right) \right] $ where $ a $ is a constant. Applying this rule in the above equation, then we will get
 $ {{C}^{'}}=5{{\left( x \right)}^{'}}+\dfrac{1}{9}{{\left( {{x}^{2}} \right)}^{'}} $
We know that $ \dfrac{d}{dx}\left( x \right)=1 $ , then we will have
 $ {{C}^{'}}=5\left( 1 \right)+\dfrac{1}{9}{{\left( {{x}^{2}} \right)}^{'}} $
We have the differentiation rule $ \dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}} $ . From the above rule we can write the value of $ {{\left( {{x}^{2}} \right)}^{'}}=2x $ . Substituting this value in the above equation, then we will get
 $ \begin{align}
  & {{C}^{'}}=5+\dfrac{1}{9}\left( 2x \right) \\
 & \Rightarrow {{C}^{'}}=\dfrac{2}{9}x+5 \\
\end{align} $
To find the minimum value of the given equation, we need to equate the derivative of the given equation to zero. Then we will have
 $ {{C}^{'}}=0 $
Substituting the value of $ {{C}^{'}}=\dfrac{2}{9}x+5 $ in the above equation, then we will get
 $ \Rightarrow \dfrac{2}{9}x+5=0 $
Simplifying the above equation, then we will get
 $ \begin{align}
  & \Rightarrow \dfrac{2}{9}x=-5 \\
 & \Rightarrow 2x=-5\times 9 \\
 & \Rightarrow 2x=-45 \\
 & \Rightarrow x=-\dfrac{45}{2} \\
 & \Rightarrow x=-22.5 \\
\end{align} $
Hence, we will get the minimum value for the function $ C=10,000+5x+\left( \dfrac{1}{9} \right){{x}^{2}} $ at $ x=-22.5 $ .

Note:
 In the question, they have only asked to calculate the value of $ x $ where the given function has a minimum value. So, we have stopped our solution after getting the value of $ x $. If they have asked to find the minimum value also then we need to substitute the calculated value of $ x $ in the given equation. Then we will get the minimum value of the function.