To find the critical values of a function, you need to find the points where the derivative of the function is either equal to zero or undefined. These points are potential locations of local maxima, local minima, or saddle points.
Here's a step-by-step guide:
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Find the Derivative: Calculate the derivative of the function, denoted as f'(x). This represents the rate of change of the function at any given point x.
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Set the Derivative Equal to Zero: Solve the equation f'(x) = 0 for x. The solutions to this equation are the points where the function has a horizontal tangent line, and they are potential critical values.
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Find Where the Derivative is Undefined: Determine any values of x where the derivative f'(x) is undefined. This often occurs when the derivative involves a fraction and the denominator is equal to zero, or when dealing with functions that have discontinuities in their derivatives (like absolute value functions or piecewise functions). These points are also considered critical values.
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Identify the Critical Values: The critical values are all the x-values found in steps 2 and 3. These are the points you'll need to analyze further to determine whether they correspond to local maxima, local minima, or neither.
Example:
Let's say you have the function f(x) = x3 - 3x2 + 2.
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Find the derivative:
f'(x) = 3x2 - 6x -
Set the derivative equal to zero:
3x2 - 6x = 0
3x(x - 2) = 0
x = 0 or x = 2 -
Find where the derivative is undefined:
The derivative f'(x) = 3x2 - 6x is defined for all real numbers, so there are no additional critical values from this step. -
Identify the critical values:
The critical values are x = 0 and x = 2.
To determine whether these critical values represent local maxima or minima, you can use the first or second derivative test.
In summary: Finding critical values using derivatives involves taking the derivative of a function, setting it equal to zero, finding where it's undefined, and solving for x. These x-values are your critical points.
