Thread: local max and local min values

Hi

The goal is to find the local max and local min values I get so far and then I don't know where to go - there are 2 questions - a and b

a) f(x) = x / (1+x)^2

f'(x) = (1-x)/(1+x)^3 according to my calculations which means that
x is underfined at -1 and f'(0) when x = 1 so

x= 1 is a critical point

f(1) = 1

but now what - is the local maximum and minimum both 1???



b) f(x) = x^3 - 3x^2 +5

f'(x) = 3x^2 - 6x
f'(x) = 3x(x-2)

so the critical points are x = 0 and x = 2

so f(0) = 5 and f(2) = 1 but how do you know which is local max and local min since I understand that sometimes the min is greater than the max

Help me understand - thanks

calculus beginner

for critical values where f(x) is defined, the following tests are used (also called the first and second derivative tests for extrema) ...

first derivative test ...

f(x) has a max if f'(x) changes sign from (+) to (-) at that critical value.

f(x) has a min if f'(x) changes sign from (-) to (+) at that critical value.


second derivative test ...

if f''(x) > 0 at the critical value, then f(x) is concave down and a max exists at that critical value.

if f''(x) < 0, then f(x) is concave up and a min exists at that critical value.