Question

can sumone teach me about the application of differentiation in increasing/decreasing functions and concavity... please sumone be my teacher... huhu....

Answer 1

If you differentiate an increasing function, the derivative will always be positive, vice versa for decreasing.

If you differentiate a concave function twice then the result is always positive, vice versa for convex.

Answer 2

in a word:
derivative is positive then function is increasing
derivative is negative then function is decreasing
second derivative is positive function is concave up (leaning left, holding water)
second derivative negative function is concave down (leaning right, spilling water)

Answer 3

ok.. can sumone give me an example?

Answer 4

\[f(x) = \frac 1x\]
\[\frac{ \delta f(x)}{\delta x} = \frac{-1}{x^2}\]
This is negative for all x>0 thus f(x) is decreasing for all x>0.
\[\frac {\delta ^2 f(x)}{\delta x^2} = \frac 2{x^3}\]
This is positive for x>0 and thus f(x) is concave for x>0

Answer 5

diffrentiate by using quotient rule right?

Answer 6

Not at all. Use the well known fact that:
\[f(x) = x ^{n}\]
\[\rightarrow \frac {\delta f(x)}{\delta x} = nx^{n-1}\]
and the fact that
\[\frac 1x = x^{-1}\]
and
\[\frac {-1}{x^2} = -x^{-2}\]

Answer 7

ouh.. for the concavity.. we need to find the interval to knw whether the concve up or down... right?

Answer 8

Well, the interval helps in this case as if x<0 then it happens to be opposite (but not in general)

Answer 9

ouh... can u shw me an example for that?