Question

How do you find all points of inflection given `y=-x^4+3x^2-4`?

Answer 1

`(1,-2) and (-1,-2)" "`are two inflection points of the given function.

Explanation:

The inflection points of a function is determined by computing
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the second derivative of `y` then solving for `color(blue)(y''=0)`.
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`y=-x^4+3x^2-4`
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`y'=-4x^3+6x`
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`y''=-12x^2+6`
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To find the inflection points we would solve the equation:
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`color(blue)(y''=0)`
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`rArr-12x^2+6=0`
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`rArr-12(x^2-1)=0`
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`rArr-12(x-1)(x+1)=0`
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`rArrx-1=0rArrx=1" "`
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Or
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`x+1=0rArrx=-1`
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The ordinate of the point of abscissa `x=1` is:
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`y_((x=1))=-1^4+3(1)^2-4=-1+3-4=-2`
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The ordinate of the point of abscissa `x=-1` is:
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`y_((x=-1))=-(-1)^4+3(-1)^2-4=-1+3-4=-2`
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Hence,
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`(1,-2) and (-1,-2)" "` are two inflection points of the given function.