Step-by-step explanation:
To find the maxima/minima of a function, we solve for the derivative of the given function
[tex]g(x) = \dfrac{x^2 - 3x + 4}{x^2 + 3x + 4}[/tex]
Recall that according to the quotient rule of differentiation, if a function g(x) is defined as a ratio of two functions f(x) and h(x), then the derivative of g(x) is
[tex]g'(x) = \dfrac{f'(x)h(x) - f(x)h'(x)}{[g(x)]^2}[/tex]
Let
[tex]f(x) = x^2 - 3x + 4 \Rightarrow f'(x) = 2x - 3[/tex]
[tex]h(x) = x^2 + 3x + 4 \Rightarrow g'(x) = 2x + 3[/tex]
The derivative g')x) is then
[tex]g'(x) = \dfrac{(2x - 3)(x^2 + 3x + 4) - (2x + 3)(x^2 - 3x + 4)}{(x^2 + 3x + 4)^2}[/tex]
Carrying out the multiplication and collecting all similar terms, we arrive at
[tex]g'(x) = \dfrac{6(x^2 - 4)}{(x^2 + 3x + 4)^2}[/tex]
The maxima/minima of the function g(x) occurs where g'(x) = 0 and this happens when the numerator of g'(x) is
[tex]x^2 - 4 = 0 \Rightarrow x = \pm2[/tex]
Look at the graph above. The blue line represents the function g(x) and the red line is for the derivative of g(x) and you can clearly see that maxima/minima occurs when the red line intersects the horizontal axis, i.e., becomes zero at [tex]x = \pm2.[/tex]
