What are the critical numbers of #f(x) = x^2*(1 + 3 ln x)#?
The only critical number is
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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}} ).
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