The leading coefficient of this quadratic function is negative, so the graph is concave down and it has a maximum. Find the axis of symmetry first.

x = -(16) / 2(-4) = -16 / -8 = 2.

Now evaluate f(2).

-4(2^2) + 16(2) + 3 = -16 + 32 + 3 = 19.

The maximum is (2, 19).

If the leading coefficient is negative ( - 4 ); Its graph is a parabola concave down. Then it graph opens downward and the function has a maximum.

The X of the max is : x = -b /2a = - 16 /-8 = 2

Replace x = 2 in the original equation. f(2) = - 4(2)^2 + 16 (2) + 3 => .... 16 + 32 + 3 = 51

ANSWER : There is a maximum at : ( 2, 51 )

f(x) = -4x^2+16x+3

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

= -4(x-2)^2+19

(x-2)^2 is always greater than or equal to zero. That implies the function will have a maximum value at some point.

to find the maximum value,

you should get the minimum possible value for the term (x-2)^2, that is equal to zero.

this gives you that when x =2 you will get a maximum value of 19 for the function. (just substitute x = 2 in the function)

Re-write f(x) in vertex form (y = a(x - h)^2 + k). Note that if a < 0, then the function will have a maximum value. If a > 0, the function will have a minimum value.

f(x) = -4x^2 + 16x + 3

f(x) = -4(x^2 - 4x - 3/4)

f(x) = -4((x - 2)^2 - 4 - 3/4)

f(x) = -4(x - 2)^2 + 16 + 3

f(x) = -4(x - 2)^2 + 19

The vertex is (h, k) or (2, 19). So the maximum is 19 at x = 2.

f(x) = -4x² + 15x + 3

It is a parabola, so will have a maximum value. We can get this by solving for the zero of the first derivative:

f'(x) = -8x + 15

0 = -8x + 15

8x = 15

x = 15/8

So the maximum occurs when x = 15/8. We can now find that value:

f(x) = -4x² + 15x + 3

f(15/8) = -4(15/8)² + 15(15/8) + 3

f(15/8) = -4(225/64) + 225/8 + 3

f(15/8) = -900/64 + 225/8 + 3

common denominators:

f(15/8) = -900/64 + 1800/64 + 192/64

f(15/8) = 1092/64

reducing the fraction:

f(15/8) = 273/16

that is the maximum value of your curve.