How do you find the critical numbers of #f(x) = (4 x - 6)e^{-6 x} #?

Answer 1

The only critical number is #5/3#

A critical number for #f# is a number, #c#, in the domain of #f# with #f'(c) = 0# or f'(c)# does not exist.
For #f(x) = (4x-6)e^(-6x)# the domain is #(-oo,oo)#.
#f'(x) = 4e^(-6x) + (4x-6)(-6e^(-6x))#
# = 4e^(-6x)-24xe^(-6x)+36e^(-6x)#
# = -24xe^(-6x)+40e^(-6x)#
# = -8e^(-6x)(3x-5)#
#f'(x)# is defined for all real #x# and #f'(x) = 0# at #x=5/3#
The only critical number is #5/3#
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Answer 2

To find the critical numbers of ( f(x) = (4x - 6)e^{-6x} ), you first need to find the derivative of the function, ( f'(x) ), and then solve for the values of ( x ) where ( f'(x) = 0 ) or ( f'(x) ) is undefined.

( f'(x) = (4 - 24x)e^{-6x} - 6(4x - 6)e^{-6x} )

Next, set ( f'(x) ) equal to zero and solve for ( x ):

( (4 - 24x)e^{-6x} - 6(4x - 6)e^{-6x} = 0 )

Solve for ( x ) in the equation above to find the critical numbers.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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