For the polynomial function \( f(x)=-4 x^{4}+8 x^{3} \), answer the parts a through e. D. The graph of \( \mathrm{f}(\mathrm{x}) \) falls to the left and rises to the right. b. Find the \( x \)-intercepts. State whether the graph crosses the \( x \)-axis, or touches the \( x \)-axis and turns around, at each intercept. The \( x \)-intercept(s) is/are 0,2 . (Type an integer or a decimal. Use a comma to separate answers as needed. Type each answer only once.) At which \( x \)-intercept(s) does the graph cross the x -axis? Select the correct choice below and tif necessary, fill in the answer box to complete your choice. A. The \( x \)-intercept(s) at which the graph crosses the \( x \)-axis is/are 0,2 . (Type an integer or a decimal. Use a comma to separate answers as needed. Type each answer only once.) B. There are no \( x \)-intercepts at which the graph crosses the \( x \)-axis. At which \( x \)-intercept(s) does the graph touch the \( x \)-axis and turn around? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The \( x \)-intercept(s) at which the graph touches the \( x \)-axis and turns around is/are \( \square \) . (Type an integer or a decimal. Use a comma to separate answers as needed. Type each answer only once.) B. There are no \( x \)-intercepts at which the graph touches the \( x \)-axis and turns around. c. Find the \( y \)-intercept. The \( y \)-intercept is \( \square \) \( \square \). (Simplify your answer. Type an integer or a decimal.)
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To find the \( x \)-intercepts of the function \( f(x) = -4x^4 + 8x^3 \), we set \( f(x) = 0 \). This gives us \( -4x^4 + 8x^3 = 0 \), which factors to \( -4x^3(x - 2) = 0 \). This leads us to the intercepts at \( x = 0 \) (of order 3, indicating it touches and turns around) and \( x = 2 \) (of order 1, indicating it crosses the x-axis). For the \( y \)-intercept, we find it by plugging in \( x = 0 \) into the function: \( f(0) = -4(0)^4 + 8(0)^3 = 0 \). Thus, the \( y \)-intercept is \( 0 \). So, summarizing: - The \( x \)-intercepts are \( 0, 2 \). - The graph crosses the \( x \)-axis at \( 2 \). - The graph touches the \( x \)-axis and turns around at \( 0 \). - The \( y \)-intercept is \( 0 \).
