Question

) Let the price and quantity functions of a product be given by p = 2x+k and q = x+2 respectively, where x is a variable and k is a constant. Find the value(s) of k for which the total revenue function is always positive.

Answer 1

k must be a whole number so that the total revenue function is always positive.

Step-by-step explanation:

We are given that the price and quantity functions of a product be given by p = 2x+k and q = x+2 respectively, where x is a variable and k is a constant.

And we have to find the value(s) of k for which the total revenue function is always positive.

Price function of a product is = p(x) = $2x +k$

Quantity function of a product is = q(x) = $x+2$

As we know that the revenue of any product is calculated by multiplying the price with the quantity of that quantity, i.e;

Total Revenue function = Price function $\times$ Quantity function

      R(x)  =  p(x) $\times$ q(x)

              =  $(2x+k) \times (x+2)$

              =  $2x^{2} +4x + kx +2k$

              =  $2x^{2} +(4 + k)x +2k$

Now, for making the total revenue function always positive, we can see that the term $(2x^{2} )$ is always positive as it is a square term and the other two terms will always be positive if the value of k will be a whole number.

This means that k must be a whole number which makes the total revenue function always positive.