How do you find the critical points #f(x)= 2x^3 + 3x^2 - 36x + 5#?

Answer 1

#(-3,86)# and #(2, -39)#

From Wikipedia: a critical point or stationary point of a differentiable function of a single real variable, #f(x)#, is a value #x_0# in the domain of #f# where its derivative is 0: #f′(x_0) = 0#
Thus, to find the critical points of #f(x) = 2x^3+3x^2-36x+5#, we first need to compute #f'(x)# then find all the #x#-values such that #f'(x)=0#.
#f(x) = 2x^3+3x^2-36x+5# #f'(x) = 6x^2+6x-36x#
When #f'(x)=0#: #6x^2+6x-36x=0# #x^2+x-6x=0# #(x+3)(x-2)=0# #x=-3# or #x=2#
#x=-3#, #y=2(-3)^3+3(-3)^2-36(-3)+5=86# #x=2#, #y=2(2)^3+3(2)^2-36(2)+5=-39#
Thus, critical points are #(-3,86)# and #(2, -39)# You could conjecture that the first is a maximum point and the second a minimum, but you'll need the second derivative test to prove that in totality.
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Answer 2

To find the critical points of ( f(x) = 2x^3 + 3x^2 - 36x + 5 ), follow these steps:

  1. Find the derivative of ( f(x) ), denoted as ( f'(x) ).
  2. Set ( f'(x) ) equal to zero and solve for ( x ).
  3. The values of ( x ) obtained in step 2 are the critical points of the function.

Let's go through these steps:

  1. Find the derivative ( f'(x) ): [ f'(x) = 6x^2 + 6x - 36 ]

  2. Set ( f'(x) ) equal to zero and solve for ( x ): [ 6x^2 + 6x - 36 = 0 ] [ x^2 + x - 6 = 0 ]

Now, solve the quadratic equation using factoring, quadratic formula, or any other method. The solutions are the critical points.

  1. Once you find the solutions for ( x ), those values represent the critical points of the function.
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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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