Find the maximum profit that a company can make, if the profit function is given by p(x) = 41−24x−18x2
Find the maximum profit that a company can make, if the profit function is given by p(x) = 41−24x−18x2
Solution and Explanation
The profit function is given as p(x) = 41−24x−18x2.
p'(x)=-24-36x
p''(x)=-36
Now,
p'(x)=0=x=-\(-\frac{24}{36}\)=\(-\frac{2}{3}\)
Also,
p'(\(-\frac{2}{3}\))=-36<0
By second derivative test,x=\(-\frac{2}{3}\) is the point of local maxima of p.
= Maximunm profit =p(\(-\frac{2}{3}\))
=41-24(\(-\frac{2}{3}\))-18(\(-\frac{2}{3}\))2
=41+16-8
=49
Hence, the maximum profit that the company can make is 49 units.
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Concepts Used:
Maxima and Minima
What are Maxima and Minima of a Function?
The extrema of a function are very well known as Maxima and minima. Maxima is the maximum and minima is the minimum value of a function within the given set of ranges.
There are two types of maxima and minima that exist in a function, such as:
- Local Maxima and Minima
- Absolute or Global Maxima and Minima