Calculus. Maximum and minimum.

Sample
shahbaz ahmed
August 2024




Second derivative test for Local Extrema.


Suppose f ” is continuous on an open interval that contain

x=c.

1. If f ’(c)=0 and f ′′ (c) < 0, then f has a local maximum at

x=c.

2. If f ’(c)=0 and f ′′ (c) > 0, then f has a local minimum at

x=c.

............................................


ln x
Show that y = x has maximum at x=e


solution


ln x
y=
x

1

, Or



xy = ln x

Taking derivative with respect to x.


d d
(xy) = ln x
dx dx
dx dy 1
y +x =
dx dx x


dy 1
y+x =
dx x
dy 1
x = −y
dx x
dy 1 − xy
x =
dx x
dy 1 − xy
= (1)
dx x2
dy
Putting dx =0


dy 1 − xy
= =0
dx x2




2