How do you find the maximum value of #y = −2x^2 − 3x + 2#?

Answer 1

The maximum value of the function is #25/8#.

We can tell two things about this function before we begin approaching the problem:

1) As #x -> -infty# or #x -> infty#, #y -> -infty#. This means our function will have an absolute maximum, as opposed to a local maximum or no maxima at all.
  1. The polynomial is of degree two, meaning it changes direction only once. Thus, the only point at which is changes direction must also be our maximum. In a higher degree polynomial, it might be necessary to compute multiple local maxima and determine which is the largest.
To find the maximum, we first find the #x# value at which the function changes direction. this will be the point where #dy/dx = 0#. #dy/dx = -4x - 3# #0 = -4x - 3# #3 = -4x# #x = -3/4#
This point must be our local maximum. The value at that point is determined by calculating the value of the function at that point: #y = -2 (-3/4)^2 - 3(-3/4) + 2# #= -18/16 + 9/4 + 2# #= -9/8 + 18/8 + 16/8# #= 25/8#
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Answer 2

To find the maximum value of ( y = -2x^2 - 3x + 2 ), you can use the vertex formula for a quadratic function. The vertex of a parabola in the form ( y = ax^2 + bx + c ) is given by:

[ x = -\frac{b}{2a} ]

Substitute the values of ( a ) and ( b ) from the equation ( y = -2x^2 - 3x + 2 ) into the vertex formula to find the x-coordinate of the vertex. Then, substitute this x-coordinate into the original equation to find the corresponding y-coordinate.

  1. Determine the values of ( a ) and ( b ): ( a = -2 ) and ( b = -3 )

  2. Apply the vertex formula to find the x-coordinate of the vertex: ( x = -\frac{b}{2a} = -\frac{-3}{2(-2)} = \frac{3}{4} )

  3. Substitute ( x = \frac{3}{4} ) into the original equation to find the corresponding y-coordinate: ( y = -2\left(\frac{3}{4}\right)^2 - 3\left(\frac{3}{4}\right) + 2 )

  4. Calculate ( y ): ( y = -2\left(\frac{9}{16}\right) - 3\left(\frac{3}{4}\right) + 2 = -\frac{9}{8} - \frac{9}{4} + 2 = -\frac{9}{8} - \frac{18}{8} + \frac{16}{8} = -\frac{11}{8} )

So, the maximum value of the function ( y = -2x^2 - 3x + 2 ) occurs when ( x = \frac{3}{4} ) and ( y = -\frac{11}{8} ).

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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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