What are the extrema of #f(x)=-x^2 +5x -1 #?

Answer 1

relative max at #(5/2, 21/4) = (2.5, 5.25)#

Find the first derivative: #f(x)' = -2x + 5#
Find the critical number(s): #f'(x) = 0; x = 5/2#

Use the 2nd derivative test to see if the critical number is a relative max. or relative min.:

#f''(x) = -2; f''(5/2) < 0#; relative max. at #x = 5/2#

Find the y-value of the maximum:

#f(5/2) = -(5/2)^2 + 5(5/2) - 1 = -25/4 + 25/2 -1 = -25/4 + 50/4 - 4/4 = 21/4#
relative max at #(5/2, 21/4) = (2.5, 5.25)#
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Answer 2

To find the extrema of ( f(x) = -x^2 + 5x - 1 ), you first need to find its critical points.

  1. Take the derivative of the function: [ f'(x) = -2x + 5 ]

  2. Set the derivative equal to zero and solve for ( x ) to find critical points: [ -2x + 5 = 0 ] [ 2x = 5 ] [ x = \frac{5}{2} ]

  3. Test the critical point to determine if it is a maximum or minimum: [ f''(x) = -2 ] Since the second derivative is negative, the critical point ( x = \frac{5}{2} ) corresponds to a maximum.

Therefore, the maximum (extrema) of the function occurs at ( x = \frac{5}{2} ). To find the maximum value, substitute ( x = \frac{5}{2} ) back into the original function: [ f\left(\frac{5}{2}\right) = -\left(\frac{5}{2}\right)^2 + 5\left(\frac{5}{2}\right) - 1 ] [ f\left(\frac{5}{2}\right) = -\frac{25}{4} + \frac{25}{2} - 1 ] [ f\left(\frac{5}{2}\right) = \frac{25}{4} - 1 ] [ f\left(\frac{5}{2}\right) = \frac{25}{4} - \frac{4}{4} ] [ f\left(\frac{5}{2}\right) = \frac{21}{4} ]

So, the maximum value (extrema) of the function ( f(x) ) is ( \frac{21}{4} ) when ( x = \frac{5}{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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