For the polynomial function \( f(x)=-4 x^{4}+8 x^{3} \), answer the parts a through e. a. Use the Leading Coefficient Test to determine the graph's end behavior. A. The graph of \( f(x) \) falls to the left and falls to the right. B. The graph of \( f(x) \) rises to the left and rises to the right. C. The graph of \( f(x) \) rises to the left and falls to the right. D. The graph of \( f(x) \) falls to the left and rises to the right. The the x-intercepts. State whether the graph crosses the \( x \)-axis, or touches the \( x \)-axis and turns around, at each intere \( \square \). (Type an integer or a decimal. Use a comma to separate answers as needed. Type each answer only once.) .

**Step 1. Determine the End Behavior** The polynomial is \[ f(x) = -4x^4 + 8x^3. \] The leading term is \(-4x^4\). Since the degree is even (\(4\)) and the leading coefficient is negative (\(-4\)), the graph falls on both ends. Thus, the answer for part (a) is: A. The graph of \( f(x) \) falls to the left and falls to the right. --- **Step 2. Find the x-Intercepts** Set \( f(x) = 0 \): \[ -4x^4 + 8x^3 = 0. \] Factor out the common term: \[ -4x^3(x - 2) = 0. \] This gives the factors: \[ -4x^3 = 0 \quad \text{and} \quad x - 2 = 0. \] Solving these, we obtain: - From \(-4x^3 = 0\), we have \( x = 0 \). (Multiplicity 3) - From \( x - 2 = 0\), we have \( x = 2 \). (Multiplicity 1) --- **Step 3. Determine Crossing or Touching at Each x-Intercept** - For \( x = 0 \): Multiplicity is 3 (an odd number), so the graph **crosses** the \( x \)-axis. - For \( x = 2 \): Multiplicity is 1 (also odd), so the graph **crosses** the \( x \)-axis. Thus, the x-intercepts are \( 0 \) and \( 2 \), and at both points the graph crosses the \( x \)-axis.