Confused Calc student

Anonymous

Question description

Consider the family of functions f(x) = ax^2 − 2 for nonzero constants a. In this problem, we will find the points on the parabola given by f(x) = ax^2 − 2 that are closest to the origin.

c) In this situation (and many others) it is easier to find the values of x that minimize the function D(x) = (d(x))2 (the square of the distance) than the values of x that minimize d(x) itself. Give a brief explanation of why the values of x that minimize d(x) and the values of x that minimize D(x) are the same in this situation.

d) Find the points on the parabola that are closest to the origin. Your answer may depend on a. (Note that in some cases there may only be one such point.) Use calculus to justify your answer, and be sure to include enough evidence to demonstrate that the points you have found do indeed minimize the distance. [Hint: The Second Derivative Test may be useful.]

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