The gradient function of (y = ax^{2} + bx + c) is (8x + 4). If the function has a minimum value of 1, find the values of a, b and c.
Explanation
(y = ax^{2} + bx + c)
Gradient = (frac{mathrm d y}{mathrm d x} = 2ax + b = 8x + 4)
Equating, we have
(2a = 8 implies a = 4)
(b = 4)
For minimum value, gradient = 0
(8x + 4 = 0 implies x = -frac{1}{2})
At (x = -frac{1}{2}, y = 1)
(1 = 4(-frac{1}{2})^{2} + 4(-frac{1}{2}) + c)
(1 = 1 – 2 + c)
(1 = c – 1 implies c = 2)
(a, b, c = 4, 4, 2).