Find the location of the local extrema of the following function.

What is/are the local minimum/minima? Select the correct choice and, if necessary, fill in the answer box to complete your choice.
A. The local minimum/minima is/are at .
(Use a comma to separate answers as needed. Type integers or simplified fractions.)
B. The function has no local minimum.
What is/are the local maximum/maxima? Select the correct choice and, if necessary, fill in the answer box to complete your choice.
A. The local maximum/maxima is/are at .
(Use a comma to separate answers as needed. Type integers or simplified fractions.)
B. The function has no local maximum.

Ask by Bond Cruz. in the United States

Mar 27,2025

Question

Find the location of the local extrema of the following function.

What is/are the local minimum/minima? Select the correct choice and, if necessary, fill in the answer box to complete your choice.
A. The local minimum/minima is/are at .
(Use a comma to separate answers as needed. Type integers or simplified fractions.)
B. The function has no local minimum.
What is/are the local maximum/maxima? Select the correct choice and, if necessary, fill in the answer box to complete your choice.
A. The local maximum/maxima is/are at .
(Use a comma to separate answers as needed. Type integers or simplified fractions.)
B. The function has no local maximum.

Ask by Bond Cruz. in the United States

Mar 27,2025

UpStudy AI · Accepted Answer

1. Write the function: $$ f(x)=11-x^{\frac{6}{11}} $$ 2. Compute the derivative using the power rule: $$ f'(x) = -\frac{6}{11}\,x^{\frac{6}{11}-1} = -\frac{6}{11}\,x^{-\frac{5}{11}}. $$ 3. Set the derivative equal to zero: $$ -\frac{6}{11}\,x^{-\frac{5}{11}} = 0. $$ The constant $$-\frac{6}{11}$$ is never zero and $$x^{-\frac{5}{11}}$$ is defined (except at $$x=0$$). Thus, there is no solution for $$f'(x)=0$$. 4. Examine where the derivative is undefined. The derivative is undefined when $$ x^{-\frac{5}{11}} $$ is undefined, which occurs at $$x=0$$. Since $$0$$ is in the domain of $$f$$ (because $$0^{\frac{6}{11}}=0$$), $$x=0$$ is a critical point. 5. Determine the sign change of $$f'(x)$$ around $$x=0$$: - For $$x>0$$: Since $$x^{-\frac{5}{11}}>0$$, $$ f'(x) = -\frac{6}{11}\,x^{-\frac{5}{11}} < 0. $$ - For $$x<0$$: Write $$ x^{\frac{5}{11}} = (\sqrt[11]{x})^5. $$ When $$x<0$$, $$\sqrt[11]{x}$$ is negative (because the 11th root of a negative number is negative) and its 5th power remains negative. Thus, $$x^{\frac{5}{11}}$$ is negative and hence $$ x^{-\frac{5}{11}} = \frac{1}{x^{\frac{5}{11}}} $$ is also negative. Therefore, $$ f'(x) = -\frac{6}{11}\,( \text{negative}) > 0. $$ 6. Conclude the behavior at $$x=0$$: - For $$x<0$$, $$f'(x)>0$$ and for $$x>0$$, $$f'(x)<0$$. This implies that the function increases to $$x=0$$ from the left and decreases from $$x=0$$ to the right, so $$x=0$$ is a local maximum. 7. Evaluate $$f(0)$$: $$ f(0)=11-0^{\frac{6}{11}}=11. $$ 8. Final answers: - Local minimum: The function has no local minimum. - Local maximum: $$f(0)=11$$ at $$x=0$$. Answer choices: Local minimum: B. The function has no local minimum. Local maximum: A. The local maximum is $$11$$ at $$x=0$$. The function has no local minimum. The local maximum is 11 at $$ x=0 $$.

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