How do you find local maximum value of f using the first and second derivative tests: #f(x) = 4x^3 + 3x^2 - 6x + 1#?

Answer 1

#f''(-1)=24*(-1)+6=(-18)# - local maximum
#f''(1/2)=24*(1/2)+6=18# - local minimum

#f'(x)=12*x^2+6*x-6# #f'(x) =0 # #12x^2+6x-6=0# #x_1=(-1)# #x_2=1/2#
there are two values -local maximum #x_1 , x_2#
#f''(x)=24x+6#
#f''(-1)=24*(-1)+6=(-18)# - local maximum, <0 #f''(1/2)=24*(1/2)+6=18# - local minimum, >0
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Answer 2

To find the local maximum value of ( f(x) = 4x^3 + 3x^2 - 6x + 1 ) using the first and second derivative tests, follow these steps:

  1. Find the first derivative of ( f(x) ) to locate critical points.
  2. Set the first derivative equal to zero and solve for ( x ) to find critical points.
  3. Find the second derivative of ( f(x) ) to determine concavity.
  4. Test the critical points found in step 2 using the second derivative test.

Let's go through these steps:

  1. ( f'(x) = 12x^2 + 6x - 6 )

  2. Setting ( f'(x) ) equal to zero and solving for ( x ): ( 12x^2 + 6x - 6 = 0 ) Factor out 6: ( 6(2x^2 + x - 1) = 0 ) Solve the quadratic equation ( 2x^2 + x - 1 = 0 ) using the quadratic formula or factoring. The solutions are ( x = -1 ) and ( x = \frac{1}{2} ).

  3. ( f''(x) = 24x + 6 )

  4. Test the critical points:

    • For ( x = -1 ): ( f''(-1) = 24(-1) + 6 = -18 < 0 ). This indicates concave down, so ( x = -1 ) is a local maximum.
    • For ( x = \frac{1}{2} ): ( f''\left(\frac{1}{2}\right) = 24\left(\frac{1}{2}\right) + 6 = 18 > 0 ). This indicates concave up, so ( x = \frac{1}{2} ) is a local minimum.

Therefore, the local maximum value of ( f(x) ) occurs at ( x = -1 ). To find the corresponding ( y )-coordinate, plug ( x = -1 ) into ( f(x) ): [ f(-1) = 4(-1)^3 + 3(-1)^2 - 6(-1) + 1 = -4 + 3 + 6 + 1 = 6 ]

So, the local maximum value of ( f(x) ) is ( 6 ) when ( x = -1 ).

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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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