Answer: [tex]C(x)=\text{ 3x}^2\text{{\text{-7x + 7 }\operatorname{\lparen}\text{opt}\imaginaryI\text{onA}\operatorname{\rparen}}}[/tex]
Explanation:[tex]\begin{gathered} C^{\prime}(x)\text{ = 6x - 7} \\ Fixed\text{ cost = \$7} \\ \\ We\text{ need to find the cost function by integrating the marginal cost function} \end{gathered}[/tex][tex]\begin{gathered} \int C^{\prime}(x)\text{ = }\int(6x\text{ - 7\rparen dx} \\ C(x)\text{ = }\int6xdx\text{ - }\int7dx \\ C(x)\text{ = }\frac{6x^2}{2}-\text{ 7x +}C \\ C(x)\text{ = 3x}^2\text{ - 7x + }C \end{gathered}[/tex]
To get the value of the constant, we will equate the cost function to zero and substitute x with the fixed cost
[tex]\begin{gathered} C(x)\text{ = 7 when x = 0} \\ 7\text{ = 3x}^2\text{ - 7x + }C \\ 7\text{ = 3\lparen0\rparen}^2\text{ -7\lparen0\rparen +C} \\ 7\text{ = C} \end{gathered}[/tex]
Substitute the value of C in the function of C(x) to get the total cost function:
[tex]C(x)\text{ = 3x}^2\text{ - 7x + 7 \lparen option A\rparen}[/tex]
