Question

Identify the relative maximum value of g(x) for the function shown below.


`g(x)=(2)/(x^2+3)`

Answer 1

The maximum value of g(x) = 2/3 at x = 0

Explanation:

* Lets find the maximum value of a function using derivative of it

- The function g(x) = 2/(x² + 3)

- 1st step use the negative power to cancel the denominator

∴ g(x) = 2(x² + 3)^-1

- 2nd use derivative of g(x) to find the value of x when g'(x) = 0

* How to make the derivative of a function

# If f(x) = a(h(x))^n, then f'(x) = an[h(x)^(n-1)](h'(x))

∵
`g(x)=2(x^(2)+3)^(-1)`

∴
`g'(x) = 2(-1)(x^(2)+3)^(-2)(2x)=-4x(x^(2)+3)^(-2)`

# Put g'(x) = 0

∴
`-4x(x^(2)+3)^(-2)=0====(-4x)/((x^(2)+3)^(2))=0`

∴
`-4x=(0)(x^(2)+3)^(2)====-4x = 0`

∴ x = 0

* The maximum value of g(x) at x = 0

- Substitute the value of x in g(x)

∴ g(0) = 2/(0 + 3) = 2/3

* The maximum value of g(x) = 2/3 at x = 0