Let D be an open subset of R", let XO E D and let f : D → R be a function. Which of the following statements is/are always true? A. If f is differentiable at xo, then f is continuous at xo . B. If all first partial derivatives of f exist on D and are continuous at xo, then f is differentiable at xo . C. If all first partial derivatives of f at xo exist, then f is differentiable at xo. D. If f is differentiable at xo, then all first partial derivatives of f are continuous at xo. E. If f is differentiable at xo, then all first partial derivatives of f at xo exist. F. If all first partial derivatives of f at xo exist, then f is continuous at xo . G. If f is continuous at xo, then all first partial derivatives off at xo exist. H. None of the above.

4. Show that if f:[0,1]→[0,1] is continuous, then f has at least one fixed point. That is, there exists an xE[0,1] so that f(x) = x.

7. Modify B(z) to construct a C function C(z) which satisfies a. C(z) = 0 if S 0.. b. C(z) = 1 if > 1. c. C'(z) > 0 if0< z < 1.

If a twice differentiable function f has exactly three zeros, at x=1, x=2 and x=3, then which of the following can we necessarily conclude? Select all that apply. O f' has at least two zeros in the interval (1,3) . O f' has exactly one zero in the interval (1,3). O f' is never zero on the interval (1, 3). f" has at least one zero in the interval (1,3). f" is never zero on the interval (1 , 3) .