Question

In Exercises 115 and 116, find the absolute maximum and minimum values of each function on the given interval.
116. y = 10x (2 - ln(x)), (0, e²]"133. Find the absolute maximum value of
f(x) = x^2 * ln(1/x)
and say where it is assumed

Accepted Answer

First, rewrite the function to a more convenient form. Given $$ f(x) = x^2 \ln\left(\frac{1}{x}\right) $$, use the logarithm property $$ \ln\left(\frac{1}{x}\right) = -\ln(x)  to $$rewrite it as $$ f(x) = x^2 (-\ln(x)) = -x^2 \ln(x) . $$Next, find the critical points by computing the derivative $$ f'(x) . $$Use the product rule on $$ -x^2 \ln(x) $$: $$ f'(x) = -\left( 2x \ln(x) + x^2 \cdot \frac{1}{x} \right) = -\left( 2x \ln(x) + x \right) . $$Set the derivative equal to zero to find critical points: $$ f'(x) = 0 \Rightarrow -\left( 2x \ln(x) + x \right) = 0 \Rightarrow 2x \ln(x) + x = 0 . $$Factor out $$ x $$: $$ x(2 \ln(x) + 1) = 0 . $$Since $$ x \u003e 0 $$, solve $$ 2 \ln(x) + 1 = 0 $$ for $$ x . $$Solve for $$ x $$ from the equation $$ 2 \ln(x) + 1 = 0 $$: $$ \ln(x) = -\frac{1}{2} $$, so $$ x = e^{-1/2} . $$This is the critical point inside the domain. Evaluate $$ f(x)  at $$the critical point $$ x = e^{-1/2} $$ and at the endpoints of the domain (if given) to determine the absolute maximum value and where it is assumed.

In Exercises 115 and 116, find the absolute maximum and minimum values of each function on the given interval.
116. y = 10x (2 - ln(x)), (0, e²]"133. Find the absolute maximum value of
f(x) = x^2 * ln(1/x)
and say where it is assumed

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Finding Critical Points

Evaluating Function at Endpoints

Properties of Logarithmic Functions in Calculus

In Exercises 115 and 116, find the absolute maximum and minimum values of each function on the given interval.116. y = 10x (2 - ln(x)), (0, e²]"133. Find the absolute maximum value off(x) = x^2 * ln(1/x)and say where it is assumed

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Finding Critical Points

Evaluating Function at Endpoints

Properties of Logarithmic Functions in Calculus

In Exercises 115 and 116, find the absolute maximum and minimum values of each function on the given interval.
116. y = 10x (2 - ln(x)), (0, e²]"133. Find the absolute maximum value of
f(x) = x^2 * ln(1/x)
and say where it is assumed