Question

Let f be a function such that f(1)=2. At each point (x,y) on the graph of f, the slope is given by dy/dx =5xy-x²-y²-5. Which of the following statements is true?

A. f has a relative maximum at x=1.
B. f has a relative minimum at x=1.
C. f has neither a relative minimum nor a relative maximum at x=1.
D. There is insufficient information to determine whether f has a relative minimum. a relative maximum, or neither at x=1.

Answer 1

There is not enough information to determine if function f has a relative maximum, relative minimum, or neither at x=1, (option D) since we only know the slope is zero at that point but lack details about the behavior of the function near x=1.

Step-by-step explanation:

The student has asked about determining whether the function f has a relative maximum, relative minimum, or neither at x=1 based on the given slope dy/dx = 5xy - x² - y² - 5 and the fact that f(1) = 2.

To answer this, we need to evaluate the slope at x = 1. Substituting x=1 and f(1)=2 into the slope formula, we get:

dy/dx at x=1 = 5(1)(2) - 1² - 2² - 5 = 10 - 1 - 4 - 5 = 0

Since the slope is zero at x=1, there might be a relative extremum at this point. To determine the type of extremum, we need to assess the sign change of the slope around x=1. We can do this by examining the second derivative or by plugging values just less and just more than 1 into the first derivative to see the sign change.

However, without further information on the values around x=1, we cannot conclusively determine whether f has a relative maximum, relative minimum, or neither at x=1. Therefore, the correct option would be D. There is insufficient information to determine whether f has a relative minimum, a relative maximum, or neither at x=1.