i just dont understand..plz help me in simple words..

if in a question it is given some f(x)=..x3 +.....any expression n then asked

find the loacal maximum and local minimum values of func f ?.

then wht to do?

why do we differential..?f'(x) , and then f''(x)?...

?

Suppose you have given the function

your first question is: find the local maximum and local minimum values of the function.

We have,

For local maximum or local minimum, we must have 

                               f'(x) = 0

Thus, x = 1 and x = 6 are the possible points of local maxima or minima. Now, we test the function at each of these points.

Differentiating equation (2) with respect to x, we get –

 

At x = 1 , we have

 

x = 1 is a point of local maxima and local maximum value of function=  

Also, at x = 6, we have

 

So, x = 6 is a point of local minima.

The local minimum value = .

 

Now, you are asking that why we find f'(x) and f"(x) in these questions?

To find the local maximum and local minimum value of the function i.e. the extreme values of the function, we find the derivative of the given function because the derivative (i.e. rate of change) at the extreme points is always zero. That's why we put the derivative equals to zero to find the extreme values (i.e. values of x at which function will have local maximum or minimum value).

Also, we know that according to second derivative test to find the local maximum or local minimum values –

If  f"(x) < 0 at x = a

then a is point of maximum and

 if  f"(x) > 0 at x = a

then a is point of minimum.

That's why we find  f"(x) and then check whether  f"(x) is positive or negative at the extreme points.

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