K ∈ R + and z ∈ [ 0 , x ] ∩ [ 0 , K ] γ &gt; 0 one defines ϕ ( z ) = ( K − z ) ( z x ) γ The maximum is attained at z = γ K γ + 1 . Question: Why is for x ≤ γ K γ + 1 : max z ϕ ( z ) = K − x And for x &gt; γ K γ + 1 max z ϕ ( z ) = ϕ ( γ K γ + 1 ) ? Best regards.

Question

K  ∈            R              +       and   z  ∈  [  0  ,  x  ]  ∩  [  0  ,  K  ]   γ  &amp;gt;  0 one defines                    ϕ        (        z        )        =        (        K        −        z        )                              (                          z              x                        )                                γ                              The maximum is attained at   z  =            γ      K              γ      +      1      . Question: Why is for   x  ≤            γ      K              γ      +      1      :                              max                      z                          ϕ        (        z        )        =        K        −        x            And for   x  &amp;gt;            γ      K              γ      +      1                                    max                      z                          ϕ        (        z        )        =        ϕ                  (                                    γ              K                                      γ              +              1                                )                 ?            Best regards.

soymmernenx · Accepted Answer

The maximumis o er   z  ∈  [  0  ,  x  ]  ∩  [  0  ,  K  ]  =  [  0  ,  min  (  x  ,  K  )  ].   ϕ is increasing on   [  0  ,  γ    K      /    (  γ  +  1  )  ] and decreasing on   [  γ    K      /    (  γ  +  1  )  ,  K  ]. If   x  ≤  γ    K      /    (  γ  +  1  ), the maximum is attained precisely at   x, and its value is   ϕ  (  x  )  =  K  −  x. If   x  &amp;gt;  γ    K      /    (  γ  +  1  ), the maximum is attained at   γ    K      /    (  γ  +  1  ).

K ∈ <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-s

Answered question

Answer & Explanation

New Questions in High school geometry