(a) Determine if the upper bound theorem identifies 4 as an upper bound for the real zeros of \( f(x) \). (b) Determine if the lower bound theorem identifies -6 as a lower bound for the real zeros of \( f(x) \). \( f(x)=2 x^{5}+x^{4}+4 x^{3}+77 x^{2}-79 x+38 \)

To check if 4 is an upper bound for the real zeros of \( f(x) \), you can evaluate \( f(4) \) and then check the synthetic division of \( f(x) \) by \( (x - 4) \). If the synthetic division gives you all non-negative values in the remainder, then 4 is indeed an upper bound. For \( f(x) \) evaluated at 4, you would find that all coefficients in the final row of the synthetic division remain non-negative, thus confirming 4 as an upper bound. Now, to determine if -6 is a lower bound for the real zeros of \( f(x) \), use the same method: substitute -6 into \( f(x) \). If the coefficients during the synthetic division by \( (x + 6) \) turn out to be alternating in sign starting from a positive value, then -6 serves as a lower bound for the real zeros. Alternatively, check if \( f(-6) \) is non-positive.