Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.


ƒ(t) = t/ t² + 1

Verified step by step guidance

1. To find the critical points of the function \( f(t) = \frac{t}{t^2 + 1} \), we first need to find its derivative \( f'(t) \).
2. Use the quotient rule for differentiation, which states that if \( f(t) = \frac{u(t)}{v(t)} \), then \( f'(t) = \frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2} \). Here, \( u(t) = t \) and \( v(t) = t^2 + 1 \).
3. Calculate \( u'(t) \) and \( v'(t) \). We have \( u'(t) = 1 \) and \( v'(t) = 2t \).
4. Substitute \( u(t) \), \( v(t) \), \( u'(t) \), and \( v'(t) \) into the quotient rule formula to find \( f'(t) = \frac{(1)(t^2 + 1) - (t)(2t)}{(t^2 + 1)^2} \). Simplify the expression to get \( f'(t) = \frac{1 - t^2}{(t^2 + 1)^2} \).
5. Set \( f'(t) = 0 \) to find the critical points. This gives \( 1 - t^2 = 0 \). Solve for \( t \) to find the critical points.

Key Concepts

Critical Points

Critical points of a function occur where its derivative is either zero or undefined. These points are essential for identifying local maxima, minima, and points of inflection. To find critical points, one typically differentiates the function and solves for the values of the variable that satisfy these conditions.

Derivative

The derivative of a function measures the rate at which the function's value changes with respect to changes in its input. It is a fundamental concept in calculus that provides information about the function's slope and behavior. For the function given, finding the derivative will help identify where the critical points are located.

Rational Functions

A rational function is a ratio of two polynomials. In the given function, ƒ(t) = t / (t² + 1), the numerator is a polynomial of degree one, and the denominator is a polynomial of degree two. Understanding the properties of rational functions, such as their domain and behavior at asymptotes, is crucial for analyzing their critical points and overall shape.