Find the values of m and b that make the following function differentiable. the piecewise function f of x equals x squared when x is less than or equal to two or mx plus b when x is greater than two

Answers:     m = 4 and b = -4======    for f to be differentiable, we need the function to be continuous and the limit 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:     the value of f(2) is 4 since we take the x² piece to evaluate it.for to exist, the left- and right-hand limits must be equal to each other    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        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    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.    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.    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