(b) f(x)=3x^6-5x^5-x^3+3x^2 Falls to the left and rises to the right Rises to the left and falls to the right Rises to the left and rises to the right Falls to the left and falls to the right

To determine the behavior of the polynomial function \( f(x) = 3x^6 - 5x^5 - x^3 + 3x^2 \) as \( x \) approaches positive and negative infinity, we need to analyze its leading term. The leading term is the term with the highest power of \( x \), which in this case is \( 3x^6 \).<br /><br />For a polynomial function, the behavior as \( x \) approaches positive or negative infinity is determined by the leading term:<br /><br />- If the leading coefficient (the coefficient of the highest power term) is positive and the degree (the highest power) is even, the function rises to the right and rises to the left.<br />- If the leading coefficient is positive and the degree is odd, the function falls to the right and rises to the left.<br />- If the leading coefficient is negative and the degree is even, the function falls to the right and falls to the left.<br />- If the leading coefficient is negative and the degree is odd, the function rises to the right and falls to the left.<br /><br />In this case, the leading term is \( 3x^6 \), where the leading coefficient is positive (3) and the degree is even (6). Therefore, the function \( f(x) = 3x^6 - 5x^5 - x^3 + 3x^2 \) rises to the right and rises to the left.<br /><br />So, the correct answer is:<br />- Rises to the left and rises to the right