The derivative of a function f is given by f'(x)= (x-3)e^x for x>0 and f(x)=7? a.) The function has a critical point at x=3. At this point, does f have a relative minimum or neither?

Answer 1

The function #f(x)# has a local maximum at #x=3#. See explanation.

To find if a function has a critical point at a place where #f'(x)=0# you have to check if the derivative changes sign at this point. If the change occurs then #f(x) has:

To check it you can calculate the second derivative:

#f''(x)=1*e^x-(x-3)e^x=(1-x+3)e^x=(2-x)e^x#

#f''(3)=(2-3)e^3=-e^3<0#

#f''(x)# is negative in #x=3#. This means that #f'(x)# is decreasing at #x=3#, this finally means that #f(x)# has a MAXIMUM at #x=3#

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Answer 2

Christopher Allers

To determine whether the function ( f ) has a relative minimum, we need to analyze the behavior of ( f' ) around ( x = 3 ).

Since ( f'(x) = (x - 3)e^x ), we first find the critical points by setting ( f'(x) ) equal to zero and solving for ( x ):

[ (x - 3)e^x = 0 ]

This equation is equal to zero when ( x = 3 ).

To determine the nature of the critical point, we use the first derivative test. For values of ( x ) greater than 3, ( f'(x) ) is positive, indicating that the function is increasing. For values of ( x ) less than 3, ( f'(x) ) is negative, indicating that the function is decreasing. Thus, at ( x = 3 ), ( f ) changes from increasing to decreasing, suggesting a relative maximum. Therefore, at ( x = 3 ), ( f ) has a relative maximum.

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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.

The derivative of a function f is given by f'(x)= (x-3)e^x for x&gt;0 and f(x)=7? a.) The function has a critical point at x=3. At this point, does f have a relative minimum or neither?

The derivative of a function f is given by f'(x)= (x-3)e^x for x>0 and f(x)=7? a.) The function has a critical point at x=3. At this point, does f have a relative minimum or neither?