What are the local extrema of #f(x)= e^(x^2)x^2e^x#?
Let's see.
Hope it Helps:)
By signing up, you agree to our Terms of Service and Privacy Policy
To find the local extrema of ( f(x) = e^{x^2}  x^2e^x ), we first need to find its critical points by setting the derivative equal to zero and solving for ( x ). Then, we can determine whether these critical points correspond to local maxima or minima by analyzing the sign of the second derivative.

Find the first derivative: [ f'(x) = (2xe^{x^2})  (2xe^x + x^2e^x) ] [ f'(x) = 2xe^{x^2}  2xe^x  x^2e^x ]

Set ( f'(x) = 0 ) and solve for ( x ): [ 2xe^{x^2}  2xe^x  x^2e^x = 0 ] [ 2xe^x(e^{x^2x}1) = 0 ]
This equation yields critical points at ( x = 0 ) and ( e^{x^2  x} = 1 ).

To analyze the nature of these critical points, we find the second derivative: [ f''(x) = (2e^{x^2} + 4x^2e^{x^2})  (2e^x + 2xe^x + 2xe^x + x^2e^x) ] [ f''(x) = 2e^{x^2}(1 + 2x^2)  2e^x(1 + 2x + x^2) ]

Evaluate ( f''(0) ) and ( f''(x) ) at the critical points where ( e^{x^2  x} = 1 ).
[ f''(0) = 2 ] [ f''(x) = 2e^{x^2}(1 + 2x^2)  2e^x(1 + 2x + x^2) ]
 Since ( f''(0) > 0 ), ( x = 0 ) corresponds to a local minimum.
For ( e^{x^2  x} = 1 ), it implies ( x^2  x = 0 ), which yields solutions ( x = 0 ) and ( x = 1 ).

Evaluate ( f''(1) ): [ f''(1) = 2e^{1^2}(1 + 2\cdot1^2)  2e^1(1 + 2\cdot1 + 1^2) ] [ f''(1) = 4e  6e ] [ f''(1) = 2e ]

Since ( f''(1) < 0 ), ( x = 1 ) corresponds to a local maximum.
Thus, the local extrema of ( f(x) = e^{x^2}  x^2e^x ) are:
 Local minimum at ( x = 0 )
 Local maximum at ( x = 1 )
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided 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 onesided 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 onesided 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 onesided 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.
 What is the derivative of #y = ln(cscx)#?
 How do I find the numbers #c# that satisfy Rolle's Theorem for #f(x)=sqrt(x)x/3# on the interval #[0,9]# ?
 What are the critical values, if any, of #f(x)= x^(3/4)  2x^(1/4)#?
 What are the absolute extrema of # f(x)= 6x^3 − 9x^2 − 36x + 3 in [4,8]#?
 What are the critical values of #f(x)=x/sqrt(x^2+2)(x1)^2#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7