How do you find the relative extrema for #f(x)=(x^4)+(4x^3)-12#?

Answer 1

#f# has global minimum at #x_0=-3# , #f(-3)=-39#

#f(x)=x^4+4x^3-12#, #D_f=RR#
#f'(x)=4x^3+12x^2=4x^2(x+3)#
#f'(x)=0 <=># #(x=0# , #x=-3)#
#f'(x)>0 <=> 4x^2(x+3)>0# #<=># #x##in##(-3,0)uu(0,+oo)#
and #f# is continuous at #x_0=0# so #f# is strictly increasing in #[-3,+oo)#
#f'(x)<0 <=> 4x^2(x+3)<0# #<=># #x##in##(-oo,-3)#
so #f# is strictly decreasing in #(-oo,-3]#
As a result #f# has relative minimum at #x_0=-3# , #f(-3)=-39# which also is an global minimum. (you can mention that #f(0)# is a saddle point.)
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Answer 2

To find the relative extrema of ( f(x) = x^4 + 4x^3 - 12 ), follow these steps:

  1. Find the derivative of the function, ( f'(x) ).
  2. Set ( f'(x) = 0 ) and solve for ( x ) to find critical points.
  3. Determine the nature of each critical point by considering the sign of the second derivative, ( f''(x) ), or by using the first derivative test.
  4. The critical points where the derivative changes sign correspond to relative extrema.

Let's proceed with these steps:

  1. Find the derivative: [ f'(x) = 4x^3 + 12x^2 ]

  2. Set ( f'(x) = 0 ) and solve for ( x ): [ 4x^3 + 12x^2 = 0 ] [ 4x^2(x + 3) = 0 ] [ x = 0 \quad \text{or} \quad x = -3 ]

  3. Determine the nature of each critical point:

    • For ( x = 0 ): ( f''(x) = 12x + 24 = 24 > 0 ), so ( x = 0 ) is a relative minimum.
    • For ( x = -3 ): ( f''(x) = 12x + 24 = 0 < 0 ), so ( x = -3 ) is a relative maximum.

Therefore, the relative minimum occurs at ( x = 0 ) and the relative maximum occurs at ( x = -3 ).

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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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