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Show that the curve y=6x^3+5x-3 has no tangent line with slope 4.

The answer key says that

m=y'=18x^2+5, but x^2 is greater than or equal to 0 for all x, so m is greater than or equal to 5 for all x.

I don't understand that x^2 is greater than or equal to zero. Where did that come from? Why isn't x all real numbers?

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1 answer
  1. y' = 18x^2 + 5

    we want 18x^2 + 5 = 4
    18x^2 = -1
    x^2 = -1/18
    but that is not possible since the square of anything has to be positive,
    so there is value of x making this equations true
    thus there is not slope of 4
    thuse there is no such tangent

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