Find the maximum value, and where it occurs for eAch function.
G(x)= (square root -x^2+4x+12)
the vertex of the parabola
y = -x^2 + x + 12
x^2 - x = -y + 12
x^2 - x + 1/4 = -y+ 12.25
(x-1/2)^2 = -1 (y-12.25)
vertex at (1/2 , 12.25)
there -x^2 + 4 x + 12.25 = -1/4+2+12.25
= 14
G = sqrt(14)
4x sort (25−x2)
nvm that is not correct! i was trying to ask a question
To find the maximum value and where it occurs for the function G(x) = √(-x^2 + 4x + 12), we need to find the vertex of the parabola formed by the function.
Step 1: Write the function in vertex form:
G(x) = √(a(x-h)^2 + k)
Step 2: Group the terms with 'x' together and complete the square:
G(x) = √(-(x^2 - 4x) + 12)
= √(-(x^2 - 4x + 4 - 4) + 12)
= √(-(x - 2)^2 + 16)
Step 3: Compare the function to the vertex form:
Here, a = -1, h = 2, and k = 16.
Step 4: The vertex of the parabola occurs at the point (h, k). Therefore, the maximum value occurs at (2, 16).
So, the maximum value of G(x) is 16, and it occurs when x = 2.