Home > Calculus > Interpreting the Sign of the First Derivative (Increasing and Decreasing Functions)

Is #f(x)=xe^x-x^2e^x # increasing or decreasing at #x=0 #?

Answer 1

#f(x)=xe^x-x^2e^x# is increasing at #x=0#.

Finding the derivative of a function at a given point indicates whether the function is increasing or decreasing; if the derivative is positive, the function is increasing, and if it is negative, it is decreasing.

As #f(x)=xe^x-x^2e^x#

#(df)/(dx)=1*e^x+x*e^x-(2xe^x+x^2e^x)# or

#(df)/(dx)=e^x+xe^x-2xe^x-x^2e^x=e^x(1-x-x^2)#

and at #x=0#, #(df)/(dx)=e^0(1-0-0^2)=1xx1=1#

as it is positive, #f(x)=xe^x-x^2e^x# is increasing at #x=0#.

graph{[-5, 5, -2.5, 2.5]} xe^x-x^2e^x

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Answer 2

Ezra Adams

To determine if ( f(x) = xe^x - x^2e^x ) is increasing or decreasing at ( x = 0 ), we can use the first derivative test.

First, find the first derivative of ( f(x) ) with respect to ( x ):

( f'(x) = (1+x)e^x - 2xe^x )

Next, evaluate ( f'(0) ):

( f'(0) = (1+0)e^0 - 2(0)e^0 = 1 )

Since ( f'(0) > 0 ), the function is increasing at ( x = 0 ).

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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.