How do you find the maximum and minimum of #y=(x+2)(x-4)#?
A quadratic equation will have either a maximum or a minimum, but not both.
This function, y = (x+2)(x - 4) has to "zeros" where the graph crosses the x-axis. Look at the picture below:
Notice that halfway in between the two zeros, the graph reaches its lowest point at the "vertex". The y-value here will be the minimum value. Let's find the y-value.
Average the two zeros:
Substitute x = 1 into the equation: y = (1+2)(1-4) =
Your vertex is at (1, -9) and the minimum value is -9.
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To find the maximum and minimum of ( y = (x+2)(x-4) ), you first need to find the critical points by taking the derivative of the function and setting it equal to zero. Then, determine the nature of these critical points by analyzing the second derivative. Finally, evaluate the function at these critical points to find the maximum and minimum values.
- Find the derivative of ( y = (x+2)(x-4) ):
[ y' = (x-4) + (x+2) = 2x - 2 ]
- Set the derivative equal to zero and solve for ( x ):
[ 2x - 2 = 0 ] [ 2x = 2 ] [ x = 1 ]
- Determine the nature of the critical point ( x = 1 ) using the second derivative test.
[ y'' = 2 ]
Since the second derivative is positive, the critical point at ( x = 1 ) corresponds to a local minimum.
- Evaluate the function at the critical point and at the endpoints of the interval:
[ y(1) = (1+2)(1-4) = -3 ] [ y(-2) = (-2+2)(-2-4) = 0 ] [ y(4) = (4+2)(4-4) = 0 ]
- Compare the values obtained. The minimum occurs at ( x = 1 ), where ( y = -3 ), and there are no maximum values.
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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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