Question

For the function g(x) = x / (x² + 1), find any critical points. On the given domains, determine if there is an absolute minimum or maximum and what the value of the extrema are.

(a) 2¹/² ≤ x ≤ 2
(b) -2 ≤ x ≤ -2¹/²
(c) 0 < x < [infinity]
(d) -[infinity] < x < [infinity]

Answer 1

To find the critical points of the function g(x) = x / (x² + 1), take the derivative of the function and set it equal to zero. The function does not have any critical points. In the given domains, there is no absolute minimum or maximum except in the domain (c) where there is an absolute minimum at x = 0.

Step-by-step explanation:

To find the critical points of the function g(x) = x / (x² + 1), we need to find the values of x where the derivative of the function is equal to zero or undefined.

1. Take the derivative of g(x) with respect to x: g'(x) = ((x² + 1)(1) - (x)(2x))/(x² + 1)².
2. Simplify the expression: g'(x) = (1 - 2x²)/(x² + 1)².
3. Set g'(x) equal to zero and solve for x: (1 - 2x²)/(x² + 1)² = 0. This equation has no real solutions.
4. Set the denominator (x² + 1)² equal to zero and solve for x. This equation also has no real solutions.

Therefore, the function g(x) does not have any critical points.

For the given domains:

• In domain (a): 2¹/² ≤ x ≤ 2, there is no absolute minimum or maximum since the function is continuous without any critical points.
• In domain (b): -2 ≤ x ≤ -2¹/², there is no absolute minimum or maximum since the function is continuous without any critical points.
• In domain (c): 0 < x < [infinity], there is an absolute minimum at x = 0 since the function approaches 0 as x approaches infinity.
• In domain (d): -[infinity] < x < [infinity], there is no absolute minimum or maximum since the function is continuous without any critical points.