Suppose that f(x, y) is a differentiable function of two variables defined on the entire xy–plane, with the following property, (i) f(x, y) is a strictly increasing function of x for each fixed value of y, and (ii) f(x, y) is a strictly increasing function of y for each fixed value of x. Let g(t) be the function of a single variable defined by g(t) = f(t, t). A. Give an example of a function f(x, y) that has the indicated property, consider the associated function g(t), and analyze its increasing/decreasing behavior. B. Make a conjecture regarding the behavior of the function g(t) in the most general setting, and then prove your conjecture

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A:

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Q: Show that the function f(z)= Im(z) is nowhere differentiable.

A:

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.