Question #182299

Let 

f

be a differentiable function on 

[,  ]

and 

x [,  ].

Show that, if 

f (x)  0

and 

f (x)  0,

then 

f

must have a local maximum at 

x.


1
Expert's answer
2021-04-20T04:37:51-0400


The given information is f(x) has a relative minimum at x = 0. Hence 


f(x)=0 at x=0f'(x)=0\text{ at } x=0


f(0)=0⇒f(0)=0


Now the given equation is:


y=f(x)+ax+by=f(x)+ax+b

To find the extremum points we differentiate and equate it to zero


y=f(x)+a⇒y ' =f' (x)+a

f(x)+a=0⇒f ' (x)+a=0


Now for the function y to have a relative minimum ar x=0 we have to have


f(0)+a=0f '(0)+a=0


a=0.⇒a=0.


Hence b can have any values as it is not involved in the derivative function and a must be equal 0 to get a relative minimum for y at x=0.


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