Find the interval(s) where the function is increasing and the interval(s) where it is decreasing

step 1 We want to identify where the function is increasing or decreasing. F of t is equal to 2t over t

Find the interval(s) where the function is increasing and...

Key Concepts

Monotonicity

Monotonicity refers to the intervals where a function is either entirely increasing or decreasing. Understanding the concept of monotonic functions is essential for analyzing the overall behavior of functions and for solving optimization problems in calculus.

Critical Points

Critical points are points in the domain of a function where the derivative is zero or undefined. These points are significant as they potentially mark transitions between increasing and decreasing intervals, and they are often used to determine local extrema.

Sign Chart Analysis

After finding the derivative and identifying the critical points, a sign chart is used to test the sign of the derivative in different intervals. This analysis helps in determining on which intervals the function is increasing (if the derivative is positive) or decreasing (if the derivative is negative).

Derivative

The derivative of a function provides the instantaneous rate of change. By calculating the derivative, one can determine where the function is increasing (positive derivative) or decreasing (negative derivative). This concept is fundamental in calculus for analyzing the behavior of functions.

Quotient Rule

When dealing with a function that is expressed as a ratio of two differentiable functions, the quotient rule is applied to compute its derivative. The quotient rule states that the derivative of a quotient is given by (denominator*derivative of numerator - numerator*derivative of denominator) divided by the square of the denominator.

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