Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. ln(x^2+1)

The minimum value of the given function is f(0) = 0

Step-by-step explanation:

Explanation:-

Extreme value :-  f(a, b) is said to be an extreme value of given function 'f' , if it is a maximum or minimum value.

i) the necessary and sufficient condition for f(x)  to have a maximum or minimum at given point.

ii)  find first derivative and equating zero

iii) solve and find 'x' values

iv) Find second derivative then find the minimum value at x=a

v) Find second derivative then find the maximum value at x=a

Problem:-

Given function is f(x) = log ( x^2 +1)

step1:- find first derivative and equating zero

 ……………(1)

the point is x=0

step2:-

Again differentiating with respective to 'x', we get

on simplification , we get

put x= 0 we get   > 0

then find the minimum value at x=0

Final answer:-

The minimum value of the given function is f(0) = 0