Evaluate \( f(x)=x^{3}-5 x^{2}+3 x+6 \) at 1 and 3 to determine if the Intermediate Value Theorem guarantees that a zero exists between the two values. a) \( f(1)=5 \) b) \( f(3)= \)

To determine whether the Intermediate Value Theorem guarantees a zero exists between \( x = 1 \) and \( x = 3 \), we need to evaluate \( f(3) \). Calculating \( f(3) \): \[ f(3) = 3^3 - 5(3^2) + 3(3) + 6 = 27 - 45 + 9 + 6 = -3 \] Now we have: a) \( f(1) = 5 \) b) \( f(3) = -3 \) Since \( f(1) = 5 \) and \( f(3) = -3 \), and the function changes sign between these two points, the Intermediate Value Theorem tells us that there is at least one zero in the interval \( (1, 3) \). This means a solution exists between those values! What an exciting discovery in the world of calculus!