Mathematics

Problem

Consider the function

h(x)=\begin{cases} m x^{2}, & x<1,\\ e^{x-1}+1, & x\ge 1. \end{cases}

Find the value of

m

such that

h

is both continuous and differentiable at

x=1

Intuitive explanation

The key is that continuity and differentiability at a boundary point both depend on matching behavior from both sides. First, continuity requires the left and right limits to equal the function value at $x=1$ . Then, differentiability adds a second constraint: the left and right derivatives must also match. You'll set up two equations from these conditions and solve for $m$ .

Intuitive explanation

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