How do you find the exact relative maximum and minimum of the polynomial function of #y=x^2 +x -1#?

Answer 1

we have a minimum at #(-1/2,-5/4)#

To find a min/max we look for values of #x# that make the derivative vanish. To determine the nature of those turning points we perform the second derivative test

We are dealing with a positive quadratic so we expect a single minimum.

We have:

# y = x^2 + x -1 #
Differentiating wrt #x# we get:
# dy/dx = 2x+1 #

For the derivative to vanish we have:

# dy/dx = 0 => 2x+1 =0 # # :. x=-1/2#
Differentiating again wrt #x# we get:
# (d^2y)/(dx)^2 = 2 #
So when #x=-1/2 => (d^2y)/(dx)^2 > 0 # confirming a minimum.
Finally, When #x=-1/2 => y= 1/4-1/2-1 = -5/4 #
Thus we have a minimum at #(-1/2,-5/4)#

graph{y = x^2 + x -1 [-10, 10, -5, 5]}

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Answer 2

To find the exact relative maximum and minimum of the polynomial function ( y = x^2 + x - 1 ), follow these steps:

  1. Find the derivative of the function ( y ) with respect to ( x ) to get ( y' ).
  2. Set ( y' ) equal to zero and solve for ( x ) to find critical points.
  3. Determine the intervals where the function is increasing or decreasing using the first derivative test.
  4. Evaluate the function ( y ) at the critical points and endpoints of the intervals to find the maximum and minimum values.

By following these steps, you can determine the exact relative maximum and minimum of the given polynomial 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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