Answer
(a) 4 is an upper bound for the real zeros of \( f(x) \).
(b) -6 is a lower bound for the real zeros of \( f(x) \).
Solution
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 \( f(x) \).
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