Answers: m = 4 and b = -4======
 = \begin{cases} x^2, & x \le 2 \\ mx + b,& x \ \textgreater \ 2 \end{cases})
for f to be differentiable, we need the function to be continuous and the limit
![\lim_{h \to 0}\big[ ( f(2+h) - f(2) ) / h \big]](/tex.php?f=\lim_{h \to 0}\big[ ( f(2+h) - f(2) ) / h \big])
has to exist (the left-hand derivative is equal to the right-hand derivative)we see the pieces break at x = 2. the pieces themselves are continuous everywhere else. so we just need to establish that we need f to be continuous at x = 2. so this limit must exist for f to be continuous at x = 2:
 = \displaystyle\lim_{x \to 2}f(x))
the value of f(2) is 4 since we take the x² piece to evaluate it.for
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to exist, the left- and right-hand limits must be equal to each other
 = \lim_{x \to 2^+}f(x))
left-hand limit we use f(x) = x² since that's for x < 2. the right-hand limit we use mx + b since that's for x > 2
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by direct substitution
we do not have enough information right now to solve this. so now we examine the derivative of.the left-hand derivative of f at x = 2 we get by taking the piece x² (since that's for x < 2) and differentiating it: f'(x) = 2x when x < 2the right-hand derivative of f at x = 2 we get by taking the piece mx + b (since that's for x < 2) and differentiating it f'(x) = m when x > 2for this function to be "differentiable," it must be differentiable at every single point in its domain. x = 2 is in the domain of f so we need to make sure the derivative exists therethis limit must exist
 - f(2)}{h})
for f'(x) to be differentiable at x = 2.for that limit to exist, the left- and right-hand limits must be equal to each other.
 - f(2)}{h} = \lim_{h \to 0^+} \frac{f(2+h) - f(2)}{h})
the left-hand limit is the left-hand derivative at x = 2. the left-hand derivative at x = 2 is f'(2) = 2(2) = 4. the right-hand limit is the right-hand derivative at x=2, which is just m.
 = m \implies m = 4)
so m has to be equal to 2.use this in the equation 4 = 2m+bwe obtained earlier to solve for b: 4 = 2(4) + b b = 4 - 8 b = -4m = 4, b = -4
