Find the maximum vertical distance between a parabola f ( x ) = 3 − x 2 and a line f ( x ) = 3 − x 2 on the interval of [ − 2 , 1 ].

Question

Find the maximum vertical distance between a parabola   f  (  x  )  =  3  −      x    2   and a line   f  (  x  )  =  3  −      x    2   on the interval of   [  −  2  ,  1  ].

kavdawg8w8 · Accepted Answer

f  (  x  )  ≥  g  (  x  )  for all   −  2  ≤  x  ≤  1 and  f  (  x  )  −  g  (  x  )  =  3  −      x    2    −  x  −  1  =      9    4    −      1    4    −  x  −      x    2    =      9    4    −            (              1        2            +      x      )        2    ≤      9    4    .The equality occurs for   x  =  −      1    2    ∈  [  −  2  ,  1  ],which says that       9    4   is the answer.

Anton Huynh · Accepted Answer

You should be thinking absolute distance:      |    f  (  x  )  −  g  (  x  )      |    =      |    g  (  x  )  −  f  (  x  )      |  If you ignore the absolute value, you should be thinking about extremum instead of minimum or maximum, and then you are fine.

Find the maximum vertical distance between a parabola f(x)=3−x^2 and a line g(x)=x+1 on the interval of [−2,1]

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