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```        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?```
7 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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Sagar Singh
B.Tech IIT Delhi

```
7 years ago
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