Chapter 12.7, Problem 45E

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a.    The planes tangent to the cylinder x² + z² = 1 in ¡³ all have the form ax + bz + c = 0.

b.    Suppose $w=xy/z$, for x > 0, y > 0, and z. > 0. A decrease in z with x and y fixed results in an increase in w.

c.    The gradient ▿F(a, b, c) lies in the plane tangent to the surface F(x, y, z) = 0 at (a, b, c).