Question

explain how to find the critical numbers of a function.
to find the critical numbers of ( f ), determine all values of ( x ) where which of the following are true? (select all that apply.)
( square f^{prime}(x) ) does not exist
( square f(x)=f^{prime}(x) )
( square f(x)=0 )
( square x = 0 )
( square f^{prime}(x)=0 )

Explanation:

Critical numbers of a function \( f(x) \) are values of \( x \) in the domain of \( f \) where either \( f^{\prime}(x)=0 \) (the derivative is zero, indicating a possible local maximum, minimum, or inflection point) or \( f^{\prime}(x) \) does not exist (such as at points of discontinuity, corner - points, or vertical tangents in the graph of the function).

• For \( f(x)=f^{\prime}(x) \), this is not a standard condition for critical numbers. For example, if \( f(x)=e^{x} \), \( f(x) = f^{\prime}(x)\), but this is just a property of the exponential function and not a general rule for critical numbers.
• \( f(x) = 0 \) gives the \( x \) - intercepts of the function \( y = f(x)\), not the critical numbers.
• \( x = 0 \) is just a specific value of \( x\) and not a general condition for critical numbers.

Answer:

• \( f^{\prime}(x)\) does not exist
• \( f^{\prime}(x)=0\)