Given the function g(x)=-3x^2+12x+5. The function has ________ (enter maximum or minimum) value at point _________ (write point as (3,5) with no space).

Answer 1

f(x) =-3x^2 +12x + 5

is a downward opening parabola due to the negative sign for the coefficient of the squared term

so there is only a maximum, global and relative or local

There is no minimum point, global, relative or local.

so the answer to the first part of the problem is "Maximum"

rewrite the polynomial iin vertex form to find the specific point

-3x^2 +12x +5

=-3(x^2 -4x )+5

complete the square

=-3(x^2 -4x +4) +5 +3(4)

= -3(x-2)^2 + 17

the vertex = max point = (2,17)

(2,17) is the maximum point

17 = -3(2)^2 +12(2) +5

= -12+24 +5

=17

take the derivaive and set it equal to zero

f'(x) = -6x +12 = 0

x = 12/6 = 2 = the x coordinate of the maximum point