(Solved): Use differentiation to find the stationary point of the following quadratic functions and determine ...

Use differentiation to find the stationary point of the following quadratic functions and determine whether it is a local maximum or a local minimum using (a) the first derivative test and (b) the second derivative test. i. \( f(x)=-3 x^{2}+6 x-20 \); ii. \( g(x)=3 x^{2}+6 x+20 \). Verify your answers by completing the square. Exercise \( 7.2 \) Find the stationary points of the following functions and determine whether they are a local maximum, a local minimum or neither of these. In each case, determine whether any of the points you have found are global. i. \( f(x)=\frac{x^{3}}{3}-2 x^{2}+3 x-15 \); ii. \( g(x)=2 x^{3}+3 x^{2}+12 x-6 \). Exercise 7.3 A firm has a monopoly on its market and so it can decide the price at which it sells its product. If it sells the product for price \( p \), then demand is given by the equation \( q=300-2 p \) where \( q \) is the amount sold. The cost of producing \( q \) is given by the function \[ \mathrm{C}(q)=30+30 q-\frac{q^{2}}{10}, \] and the revenue is given by the function \( \mathrm{R}(q)=p q \). i. Find the revenue function, \( \mathrm{R}(q) \), in terms of \( q \) and hence find the profit function, \( \pi(q) \), in terms of \( q \). ii. Calculate the value of \( q \) that will give the firm its maximum profit making sure that you check that this value of \( q \) does indeed give you the maximum profit. What is the maximum profit that the firm makes and what price, \( p \), will provide this? iii. If the firm can produce at most 120 units, what price will maximise the profit?