Lecture Note
University
Massachusetts Institute of TechnologyCourse
Multivariable CalculusPages
2
Academic year
2022
Sporkz
Views
11
Comparison to the quadratic formula It needs to be pointed out: Here we have a special quadratic expression. Thequadratic formula is similar, but uses b2 – 4ac rather than b2 – 4bc. Let's see howthe quadratic formula applies here. Let`s present the same data differently to reachthe same conclusion. Here, we have written the expression as y2 times a x over y2 plus b x over y + c.Since x and y are both non-negative, the expression will never be non-negative. Thisdepends on the ratio between x and y, which means it depends on which directionyou're moving away from the origin--the point (0, 0). If a discriminant is positive, thenit means that this equation has a solution.One way to convince yourself that a quantity can be both positive and negative is toplot its square root. The square root might look like this, or it might look like that,depending on the sign of a. But, in either case, it will take values of both signs. Sothat means we have a saddle point.
While the overall situation is indicated by the quadratic equation b squared minus4ac, this equation never takes the value 0 because if b squared minus 4ac isnegative, then that means there's no solution for the quadratic equation and thus nominimum or maximum. And we know that a x over y squared plus b x over y plus c iseither always positive or always negative depending on the sign of a, which tells usthat our function will be either always positive or always negative. We'll have a maximum or a minimum, depending on whether the discriminant ispositive or negative. This will tell us whether this quadratic quantity has always thesame sign or whether I can cross the value zero when you have a root of aquadratic.
Comparison to the Quadratic Formula (1)
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