Answer:
A current of 5 amperes will produce the maximum power.
Step-by-step explanation:
Let be [tex]p(x) = -12\cdot x^{2}+120\cdot x[/tex], where [tex]p(x)[/tex] is measured in watts and [tex]x[/tex] in amperes. At first we must obtain the first and second derivatives of the function to determine the current associated with maximum power. That is:
First derivative
[tex]p'(x) = -24\cdot x + 120[/tex]
Second derivative
[tex]p''(x) = -24\cdot x[/tex]
Now, we equalize the first derivative to zero and solve it afterwards: (First Derivative Test)
[tex]-24\cdot x + 120 = 0[/tex]
[tex]x = 5\,A[/tex]
The only critical point is [tex]x = 5\,A[/tex].
As next step we need to assure that critical point leads to an absolute maximum by evaluating the critical point found above in the second derivative: (Second Derivative Test)
[tex]p(5)'' = -24\cdot (5)[/tex]
[tex]p''(5) = -120[/tex]
Which indicates that critical point leads to an absolute maximum.
A current of 5 amperes will produce the maximum power.
