Let f : R - R be a function. Note: R denotes the set of real numbers. The function is defined as:

f(x) = -x, ...

Question

Let f : R -> R be a function. Note: R denotes the set of real numbers. The function is defined as: f(x) = -x, if x < -2 f(x) = ax^2 + bx + c, if -2 <= x <= 2 f(x) = c, if x > 2 Which ONE of the following choices gives the values of a, b, c that make the function f continuous and differentiable? A. a = 1/2, b = 0, c = -1 B. b = 1/2, a = 0, c = 0 C. a = 1/2, b = -1, c = 0 D. a = 1/2, b = 1, c = -4

Accepted Answer

A. a = 1/2, b = 0, c = -1 For a piecewise function to be smooth (continuous and differentiable) at each joining point, two conditions must hold at each boundary: the function values must match (continuity) and the slopes must match (differentiability). At x = -2 we match the linear piece -x with the quadratic ax^2+bx+c. At x = 2 we match the quadratic with the constant c. This gives us a system of four equations in three unknowns (a, b, c), which the correct option satisfies exactly.