What are the local extrema, if any, of #f (x) = x^3-12x+2 #?

Answer 1

The function has 2 extrema:

#f_{max}(-2)=18# and #f_{min}(2)=-14#

We have a function: #f(x)=x^3-12x+2#

To find extrema we calculate derivative

#f'(x)=3x^2-12#
The first condition to find extreme points is that such points exist only where #f'(x)=0#
#3x^2-12=0#
#3(x^2-4)=0)#
#3(x-2)(x+2)=0#
#x=2 vv x=-2#

Now we have to check if the derivative changes sign at the calcolated points:

graph{x^2-4 [-10, 10, -4.96, 13.06]}

From the graph we can see that #f(x)# has maximum for #x=-2# and minimum for #x=2#.
Final step is to calculate the values #f(-2)# and #f(2)#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the local extrema of ( f(x) = x^3 - 12x + 2 ), we first need to find the critical points by setting the derivative equal to zero and solving for ( x ). Then we can determine if these critical points correspond to local maxima or minima.

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

  2. Set ( f'(x) ) equal to zero and solve for ( x ): [ 3x^2 - 12 = 0 ] [ x^2 - 4 = 0 ] [ x = \pm 2 ]

  3. Determine the nature of the critical points:

  • To the left of ( x = -2 ), ( f'(x) ) changes from negative to positive, indicating a local minimum.
  • To the right of ( x = -2 ), ( f'(x) ) changes from positive to negative, indicating a local maximum.
  • To the left of ( x = 2 ), ( f'(x) ) changes from negative to positive, indicating a local minimum.
  • To the right of ( x = 2 ), ( f'(x) ) changes from positive to negative, indicating a local maximum.

Therefore, the local extrema of ( f(x) ) are:

  • Local minimum at ( x = -2 )
  • Local maximum at ( x = 2 )
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7