Find f^'(c) if it exists. f(x)={ -(1)/(2) x^2, x<3 ...

Key Concepts

Piecewise Functions

Piecewise functions are defined by different formulas over different parts of their domain. When working with these functions, special attention is needed at the points where the definition changes, since the behavior of the function (such as continuity and differentiability) may differ on either side of the point.

Continuity

Continuity at a point requires that the function’s value corresponds to the limit of the function as the variable approaches that point from both sides. For piecewise functions, checking continuity at the boundary involves ensuring that the left-hand limit and right-hand limit at that point are equal and that they agree with the function's value at that point.

Differentiability

Differentiability at a point means that the function has a well-defined tangent at that point, which requires that the derivative exists. For piecewise functions, this typically entails confirming that not only is the function continuous at the point in question, but also that the derivatives (slopes) from the left and right sides match.

One-Sided Derivatives

One-sided derivatives refer to the derivatives computed as the limit of the difference quotient from either the left or the right of a point. For piecewise functions, the left-hand derivative is computed using the formula valid for values less than the point, and the right-hand derivative is computed using the formula valid for values greater than or equal to the point. The function is differentiable at the point only if these one-sided derivatives exist and are equal.

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