If f(x) is continuous and differentiable and f(x)={ a x^4+5 x ; x ≤ 2 b x^2-3 x ; x>2 .

step 1 The first thing I did is I plugged x equals 2 into both pieces of the piecewise equation. We hav

If f(x) is continuous and differentiable and f(x)={ a...

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Key Concepts

Piecewise Function

A piecewise function is defined by different expressions over different intervals of the domain. Its overall behavior depends on the rules assigned to each subdomain, and special attention is required at the boundary points where these pieces meet.

Continuity

Continuity of a function at a point means that the limit of the function from both sides equals the function’s value at that point. In the context of piecewise functions, ensuring continuity involves matching the function values at the points where the definition changes.

Differentiability

Differentiability at a point means the function has a well-defined derivative at that point, which commonly requires the left-hand and right-hand derivatives to be equal. For piecewise functions, the derivative’s behavior must be checked at the boundary to ensure there is no 'corner' or 'cusp'.

Boundary Matching Conditions

To ensure that a piecewise function is both continuous and differentiable at the boundary where the pieces meet, one must impose matching conditions. These include equating the function values for continuity and equating the derivatives from each side for differentiability, which often leads to equations that determine any unknown parameters in the function definitions.

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