Answer:
The function has a maximum in [tex]x=3[/tex]
The maximum is:
[tex]f(3) = 39[/tex]
Explanation:
Find the first derivative of the function for the inflection point, then equal to zero and solve for x
[tex]f(x)' = -4*2x + 24=0[/tex]
[tex]-4*2x + 24=0[/tex]
[tex]8x=24[/tex]
[tex]x=3[/tex]
Now find the second derivative of the function and evaluate at x = 3.
If [tex]f (3) ''< 0[/tex] the function has a maximum
If [tex]f (3) '' >0[/tex] the function has a minimum
[tex]f(x)''= 8[/tex]
Note that:
[tex]f(3)''= -8<0[/tex]
the function has a maximum in [tex]x=3[/tex]
The maximum is:
[tex]f(3)=-4(3)^2+24(3) + 3\\\\f(3) = 39[/tex]
