What are the critical values of #f(x)=(3-x)^2-x#?

Answer 1

#x = 7/2#

Given: #f(x) = (3-x)^2 - x#
You find critical values by setting #f'(x) = 0# and solving for #x#.
Find the first derivative by using the power rule : #(u^n)' = n u^(n-1)u'#
Let #u = 3-x; " "u' = -1; " " n = 2#
#f'(x) = 2(3-x)(-1) - 1#
#f'(x) = -(6-2x) - 1#
#f'(x) = -6 + 2x - 1 = 2x - 7#
#f'(x) = 2x - 7 = 0#
Critical value: #" "2x = 7; " "x = 7/2#

Find the first derivative by using distribution :

#f(x) = (3-x)(3-x) - x#
#f(x) = 9 - 6x + x^2 - x#
#f(x) = x^2 -7x + 9#
#f'(x) = 2x - 7 = 0#
Critical value: #" "2x = 7; " "x = 7/2#
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Answer 2

To find the critical values of ( f(x) = (3 - x)^2 - x ), we need to first find the derivative of the function and then set it equal to zero to solve for critical points. After finding the critical points, we can determine whether they correspond to maximum, minimum, or inflection points.

  1. Calculate the derivative of ( f(x) ) with respect to ( x ): [ f'(x) = 2(3 - x)(-1) - 1 ]

  2. Simplify the derivative: [ f'(x) = -2(3 - x) - 1 ] [ f'(x) = -6 + 2x - 1 ] [ f'(x) = -7 + 2x ]

  3. Set the derivative equal to zero and solve for ( x ) to find critical points: [ -7 + 2x = 0 ] [ 2x = 7 ] [ x = \frac{7}{2} ]

  4. Check for the nature of the critical point by examining the sign of the derivative around ( x = \frac{7}{2} ):

    • When ( x < \frac{7}{2} ), ( f'(x) < 0 ), indicating the function is decreasing.
    • When ( x > \frac{7}{2} ), ( f'(x) > 0 ), indicating the function is increasing.

Thus, the critical value of ( f(x) ) is ( x = \frac{7}{2} ).

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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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