Answer to Question #182299 in Real Analysis for Aslam

Question

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.

Answer 1

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

$f'(x)=0\text{ at } x=0$

$⇒f(0)=0$

Now the given equation is:

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

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

$⇒y ' =f' (x)+a$

$⇒f ' (x)+a=0$

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

$f '(0)+a=0$

$⇒a=0.$

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.