Question

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).ƒ(x) = x³ ln x on (0, ∞)

Accepted Answer

To find the critical points of the function $$ f(x) = x^3 \ln x  on $$the interval $$ (0, \infty) $$, first find the derivative $$ f'(x) . $$Use the product rule: if $$ u(x) = x^3 $$ and $$ v(x) = \ln x $$, then $$ f'(x) = u'(x)v(x) + u(x)v'(x) . $$Calculate the derivatives: $$ u'(x) = 3x^2 $$ and $$ v'(x) = \frac{1}{x} . $$Substitute these into the product rule to get $$ f'(x) = 3x^2 \ln x + x^3 \cdot \frac{1}{x} . $$Simplify this expression. Set $$ f'(x) = 0  to $$find the critical points. This gives the equation $$ 3x^2 \ln x + x^2 = 0 . $$Factor out $$ x^2  to $$get $$ x^2(3 \ln x + 1) = 0 . $$Since $$ x^2 
eq 0 $$ for $$ x \u003e 0 $$, solve $$ 3 \ln x + 1 = 0 . $$Solve $$ 3 \ln x + 1 = 0  to $$find the critical points. This simplifies to $$ \ln x = -\frac{1}{3} . $$Exponentiate both sides to solve for $$ x $$, giving $$ x = e^{-\frac{1}{3}} . To $$identify the absolute maximum and minimum values, evaluate $$ f(x)  at $$the critical point $$ x = e^{-\frac{1}{3}} $$ and consider the behavior of $$ f(x)  as$$ $$ x 	o 0^+ $$ and $$ x 	o \infty . $$Compare these values to determine the absolute extrema on the interval $$ (0, \infty) .$$

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
ƒ(x) = x³ ln x on (0, ∞)

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Key Concepts

Critical Points

Absolute Maximum and Minimum

Natural Logarithm Function