a. No, f is not continuous at x=4 because the left-hand limit and the right-hand limit of f(x) at x=4 are not equal. The left-hand limit of f(x) at x=4 is sqrt(4) = 2, while the right-hand limit of f(x) at x=4 is 6-4 = 2.
b. The average rate of change of f(x) on the closed interval [0,6] is (f(6)-f(0))/(6-0) = (6-sqrt(0))/6 = (6-0)/6 = 1.
c. To make g differentiable at x=4, the left-hand limit and the right-hand limit of g(x) at x=4 must be equal. This means that k sqrt(4) = 4m + 2. Solving for k and m, we get k = (4m+2)/sqrt(4) and m = (k sqrt(4)-2)/4.
Let f be the function defined by f(x)= sqrt(x), 0 <or= x <or= 4. and f(x)= 6-x, 4 < x <or= 6
a. Is f continuous at x=4? justify.
b. Find the average rate of change of f(x) on the closed interval [0,6].
c. suppose the function g(x)= k sqrt(x), 0 <or= x <or= 4. and g(x)= mx+2, 4 < x <or= 6. where k and m are constants. find the values of k and m to make g differentiable at x=4? justify.
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