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#
