# Confused Calc student

**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.]

## Tutor Answer

Brown University

1271 Tutors

California Institute of Technology

2131 Tutors

Carnegie Mellon University

982 Tutors

Columbia University

1256 Tutors

Dartmouth University

2113 Tutors

Emory University

2279 Tutors

Harvard University

599 Tutors

Massachusetts Institute of Technology

2319 Tutors

New York University

1645 Tutors

Notre Dam University

1911 Tutors

Oklahoma University

2122 Tutors

Pennsylvania State University

932 Tutors

Princeton University

1211 Tutors

Stanford University

983 Tutors

University of California

1282 Tutors

Oxford University

123 Tutors

Yale University

2325 Tutors