Given:
color(red)(f(x) = x/(x^2 + 25))
Obviously, the domain of this function is: color(blue)((-oo < x < +oo) and our function is defined.
Critical Points are points where the function is defined and its derivative is zero or undefined
color(green)(Step.1)
We have,
color(blue)(y=f(x) = x/(x^2 + 25))
We will now differentiate our f(x)
i.e., find d/(dx) [x/(x^2 + 25)]
Quotient Rule is used to differentiate.
Quotient Rule is given by
color(blue)([(u(x))/(v(x))]^' = (u'(x).v(x) - u(x)*v'(x))/(v(x)^2))
d/(dx) [x/(x^2 + 25)]
rArr [d/(dx)(x)*(x^2+25) - x*d/(dx)(x^2+25)]/(x^2+25)^2
rArr [1*(x^2+25)-{d/(dx)(x^2)+d/(dx)(25)}*x]/(x^2+25)^2
rArr (x^2 - 2x^2+25)/(x^2+25)^2
rArr ( - x^2+25)/(x^2+25)^2
Hence,
color(brown)(f'(x) = ( - x^2+25)/(x^2+25)^2)
color(green)(Step.2)
Set
color(brown)(f'(x) = 0
Hence,
color(brown)(f'(x) = ( - x^2+25)/(x^2+25)^2 = 0)
For a rational function, the derivative will be equal to zero, if the expression in the numerator is equal to zero
Set,
-x^2 + 25 = 0
Add color(red)(-25) to both sides of the equation to get
-x^2 + 25+ color(red)((-25)) = 0+ color(red)((-25)
-x^2 + cancel 25+ color(red)((- cancel 25)) = 0+ color(red)((-25)
-x^2 =-25
Divide both sides by color(red)((-1)
(-x^2)/color(red)((-1)) =-25/color(red)((-1)
x^2 =25
:. x = +-5
x = + and x = -5 are also on the domain of our function
Required Critical Points are: color(blue)(x=+5, x=-5
