Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( y=f(x) \). \( f(x)=18 x(x-1)^{3} \) What is/are the local maximum/a? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The local maximum/a is/are at \( x=\square \) B. There is no local maximum What is/are the local minimum/a? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. 10 . Use a comma to separate answers as needed.) A. The local minimum/a is/are at \( x= \). (Type an integer or simplified fraction. Use a comma to separate answers as needed.) B. There is no local minimum.
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To analyze the function \( f(x)=18 x(x-1)^{3} \), we need to find its critical points by taking the derivative and setting it to zero. The derivative \( f'(x) \) reveals that it has roots at \( x = 0 \) and \( x = 1 \). Evaluating the function, we find that \( x=0 \) represents a local maximum, yielding the point \( f(0) = 0 \), while \( x=1 \) is a local minimum since \( f(1) = 0 \). A. The local maximum occurs at \( x=0 \). A. The local minimum occurs at \( x=1 \). Drawing the graph of \( y=f(x) \), you'll see it is a cubic function that dips down to a local minimum at \( x=1 \) and rises back through the origin, showcasing the characteristic “S” shape of cubic functions.
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