Answer: To make the piecewise function continuous for all x, we need to make sure that the function values and the function slopes at the boundaries (-1 and 2) are the same for each piece.
For x = -1:
F(-1) = -6 (from the first piece)
F(-1) = A(-1) + 2 = -A + 2 (from the second piece)
So, to make the function continuous at x = -1, we need to set -6 = -A + 2, which gives us A = -8.
For x = 2:
F(2) = -x^2 + Bx + 14 (from the third piece)
F(2) = A(2) + 2 = 2A + 2 (from the second piece)
So, to make the function continuous at x = 2, we need to set -4 + 2B + 14 = 2A + 2, which gives us B = -5.
Now we can write the function with the constants that we found :
F(x) = -6 if x < -1
-8x + 2 if -1 ≤ x ≤ 2
-x^2 - 5x + 14 if x > 2
This function is continuous for all x and the values and slopes match at x = -1 and x = 2.
Explanation: