(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 determine if the upper bound theorem identifies 4 as an upper bound for the real zeros of \( f(x) \) and if the lower bound theorem identifies -6 as a lower bound for the real zeros of \( f(x) \), we need to analyze the function \( f(x) = 2x^5 + x^4 + 4x^3 + 77x^2 - 79x + 38 \). (a) To determine if 4 is an upper bound for the real zeros of \( f(x) \), we need to check if \( f(4) \) is greater than 0. If \( f(4) > 0 \), then 4 is an upper bound for the real zeros of \( f(x) \). (b) To determine if -6 is a lower bound for the real zeros of \( f(x) \), we need to check if \( f(-6) \) is less than 0. If \( f(-6) < 0 \), then -6 is a lower bound for the real zeros of \( f(x) \). Let's calculate \( f(4) \) and \( f(-6) \) to determine if they are upper and lower bounds for the real zeros of \( f(x) \). Calculate the value by following steps: - step0: Calculate: \(2\times 4^{5}+4^{4}+4\times 4^{3}+77\times 4^{2}-79\times 4+38\) - step1: Multiply the terms: \(2048+4^{4}+4\times 4^{3}+77\times 4^{2}-79\times 4+38\) - step2: Calculate: \(2048+4^{4}+4^{4}+77\times 4^{2}-79\times 4+38\) - step3: Multiply the terms: \(2048+4^{4}+4^{4}+1232-79\times 4+38\) - step4: Multiply the numbers: \(2048+4^{4}+4^{4}+1232-316+38\) - step5: Add the numbers: \(2048+512+1232-316+38\) - step6: Calculate: \(3514\) Calculate or simplify the expression \( 2*(-6)^5+(-6)^4+4*(-6)^3+77*(-6)^2-79*(-6)+38 \). Calculate the value by following steps: - step0: Calculate: \(2\left(-6\right)^{5}+\left(-6\right)^{4}+4\left(-6\right)^{3}+77\left(-6\right)^{2}-79\left(-6\right)+38\) - step1: Multiply the terms: \(-15552+\left(-6\right)^{4}+4\left(-6\right)^{3}+77\left(-6\right)^{2}-79\left(-6\right)+38\) - step2: Multiply the terms: \(-15552+\left(-6\right)^{4}-864+77\left(-6\right)^{2}-79\left(-6\right)+38\) - step3: Multiply the terms: \(-15552+\left(-6\right)^{4}-864+2772-79\left(-6\right)+38\) - step4: Multiply the numbers: \(-15552+\left(-6\right)^{4}-864+2772+474+38\) - step5: Evaluate the power: \(-15552+1296-864+2772+474+38\) - step6: Calculate: \(-11836\) (a) Since \( f(4) = 3514 > 0 \), the upper bound theorem identifies 4 as an upper bound for the real zeros of \( f(x) \). (b) Since \( f(-6) = -11836 < 0 \), the lower bound theorem identifies -6 as a lower bound for the real zeros of \( f(x) \).