Finding the maximum of G ( x ) = ∫ x x + a f ( t ) d t using FTOC.

Question

Finding the maximum of   G  (  x  )  =      ∫          x              x      +      a        f  (  t  )    d  t using FTOC.

Immanuel Glenn · Accepted Answer

If your   f is continuous then your   G is differentiable. Then applying the Leibnitz rule of differentiating under integral you calculate       G    ′    (  x  ) equate it to zero and then use the double derivative test to find out the point of extrema. Given your the function, we get      G    ′    (  x  )  =  0  +  (  −  (  x  +  a      )    2    )  −  (  −      x    2    )  =  −  (  x  +  a      )    2    +      x    2    =  −      a    2    −  2  a  x  =  0    ⟹    x  =                    −        a            2

lilmoore11p8 · Accepted Answer

Let   u  =  x  +  a so that       d          d      x       can be interchanged with       d          d      u      , so:      d          d      x            (          ∫              q                    x        +        a              −          t      2            d    t    )    =      d          d      u            (          ∫              q                    u              −          t      2            d    t    )    =  −      u    2    =  −  (  x  +  a      )    2  However, that last term should be       x    2    −  (      x    2    +  2  a  x  +      a    2    ) and then setting this to   0 gives the correct answer of   x  =            −      a        2    ..

Finding the maximum of G ( x ) = <msubsup> ∫ <mrow cla

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