Question

Consider f(x)=3ax−x2. write down a formula in terms of a for the maximum of f(x).

Answer 1

For this case we have the following function:
 $f (x) = 3ax-x ^ 2$
 To find the maximum of the function, what we should do is to derive the equation.
 We have then:
 $f '(x) = 3a-2x$
 We match zero:
 $3a-2x = 0$
 We clear the value of x:
 $2x = 3a x = (2/3) a$
 We substitute the value of x in the function to find the maximum:
 $f ((2/3) a) = 3a ((2/3) a) - ((2/3) a) ^ 2$
 Rewriting:
 $f ((2/3) a) = 2a ^ 2 - (4/9) a ^ 2 f ((2/3) a) = (18/9) a ^ 2 - (4/9) a ^ 2 f ((2/3) a) = (14/9) a ^ 2$
 Answer:
 a formula in terms of a for the maximum of f (x) is:
 $f ((2/3) a) = (14/9) a ^ 2$