Home > Calculus > Identifying Stationary Points (Critical Points) for a Function

What are the critical numbers of #f(x) = x^2*(1 + 3 ln x)#?

Answer 1

Scarlett Allison

The only critical number is #x = e^(-5/6)#

#f'(x) = 2x(1+3lnx)+x^2(3/x)= x(6lnx+5)#

#f'(x)# is defined for all #x# in the domain of #f# and

#f'(x)=0# at #lnx = -5/6# so #x = e^(-5/6)#.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Christopher Agresta

To find the critical numbers of ( f(x) = x^2(1 + 3 \ln x) ), we need to find where the derivative of ( f(x) ) equals zero or is undefined.

Take the derivative of ( f(x) ): [ f'(x) = 2x(1 + 3 \ln x) + x^2 \left(\frac{3}{x}\right) ]
Simplify the derivative: [ f'(x) = 2x + 3x \ln x + 3x ]
Set ( f'(x) ) equal to zero and solve for ( x ): [ 2x + 3x \ln x + 3x = 0 ] [ x(2 + 3 \ln x + 3) = 0 ] [ x(5 + 3 \ln x) = 0 ]

This equation gives us the critical numbers. So, ( x = 0 ) or ( 5 + 3 \ln x = 0 ).

Solving ( 5 + 3 \ln x = 0 ): [ 3 \ln x = -5 ] [ \ln x = -\frac{5}{3} ] [ x = e^{-\frac{5}{3}} ]

Therefore, the critical numbers are ( x = 0 ) and ( x = e^{-\frac{5}{3}} ).

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.