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vikash chandola Grade:
        

f(x) is a real valued function then


if f''(a)>0 then function is said to be decresing at x=a


if f''(x)<0 then function is said to be increasing at x=a


from where it comes?

6 years ago

Answers : (1)

SAGAR SINGH - IIT DELHI
879 Points
										

Dear vikash,


Let f be continuous on [a, b] and differentiable on the open interval (a, b). Then


(a) f is increasing on [a, b] if f '(x) > 0 for each x (a, b)


(b) f is decreasing on [a, b] if f '(x) < 0 for each x (a, b)


This theorem can be proved by using Mean Value Theorem. We shall prove the theorem after learning Mean Value Theorem.


This theorem is applied in various problems to check whether a function is increasing or decreasing.


(1) Let the given function be f (x) on the real number line R.


(2) Differentiate the function f(x) with respect to x and equate it to zero i.e., put f '(x) = 0. Solve for x. These values of x which satisfy f '(x) = 0 are called Critical values of the function


(3) Arrange these Critical values in ascending order and partition the domain of f (x) into various intervals, using the Critical values.


(4) Check the sign of f '(x) in each open intervals.


(5) If f '(x) > 0 in a particular interval, then the function is increasing in that particular interval.


If f '(x) < 0 in a particular interval, then the function is decreasing in that particular interval.


 










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All the best.


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Sagar Singh


B.Tech IIT Delhi









6 years ago
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