How to find range of function by differentiation?

Differentiation can be used to find the range of a function by identifying its local maxima and minima, and analyzing its behavior as x approaches infinity or negative infinity. Here's a breakdown of the process:

Steps to Find the Range of a Function Using Differentiation:

1. Find the Derivative: Calculate the first derivative, f'(x), of the function f(x).

2. Find Critical Points: Set f'(x) = 0 and solve for x. These are the critical points of the function. Also, find where f'(x) is undefined, as these points can also be local extrema.

3. Determine Intervals of Increase and Decrease: Use the critical points to divide the domain of the function into intervals. Test a value within each interval in f'(x) to determine if the function is increasing (f'(x) > 0) or decreasing (f'(x) < 0) in that interval.

4. Identify Local Maxima and Minima:

   • If f'(x) changes from positive to negative at a critical point, that point corresponds to a local maximum.
   • If f'(x) changes from negative to positive at a critical point, that point corresponds to a local minimum.
   • Calculate the y-values of these local extrema by plugging the x-values back into the original function, f(x).

5. Check End Behavior: Determine the behavior of the function as x approaches positive and negative infinity. This helps identify any horizontal asymptotes or unbounded behavior.

6. Determine the Range: Based on the local extrema and end behavior, determine the set of all possible output values of the function.

Example:

Let's find the range of the function f(x) = x³ - 3x² + 1.

1. Find the Derivative:

   • f'(x) = 3x² - 6x

2. Find Critical Points:

   • Set f'(x) = 0:
     • 3x² - 6x = 0
     • 3x(x - 2) = 0
     • x = 0, x = 2

3. Determine Intervals of Increase and Decrease:

   Interval	Test Value	f'(x) Sign	Function Behavior
   x < 0	x = -1	Positive	Increasing
   0 < x < 2	x = 1	Negative	Decreasing
   x > 2	x = 3	Positive	Increasing

4. Identify Local Maxima and Minima:

   • At x = 0, f'(x) changes from positive to negative, so there's a local maximum. f(0) = 1.
   • At x = 2, f'(x) changes from negative to positive, so there's a local minimum. f(2) = 2³ - 3(2)² + 1 = 8 - 12 + 1 = -3.

5. Check End Behavior:

   • As x → ∞, f(x) → ∞.
   • As x → -∞, f(x) → -∞.

6. Determine the Range: Since the function goes to infinity in both directions, and we have a local maximum at 1 and a local minimum at -3, the range of the function is (-∞, ∞).

Logarithmic Differentiation

The provided reference mentions logarithmic differentiation. While not directly for finding the range, it's a technique used to differentiate complicated functions, often involving products, quotients, and exponents.

Logarithmic differentiation involves the following steps (as outlined in the reference):

1. Take the natural logarithm of both sides of the equation y = f(x).
2. Use properties of logarithms to simplify the equation.
3. Differentiate both sides with respect to x using implicit differentiation. This technique is particularly useful when dealing with functions of the form y = f(x)^g(x).

While logarithmic differentiation helps in finding the derivative, it is not directly used to find the range. You still need to apply the steps outlined above (finding critical points, intervals of increase/decrease, etc.) after finding the derivative using logarithmic differentiation.