The table below shows certain values of k(x) on the interval [–17,–5].x k(x)–17 –11–13 –5–9 10–5 –14If k(x) is continuous on [–17,–5], can you conclude that the equation k(x)=–4 has a solution?
Question
The table below shows certain values of k(x) on the interval [–17,–5].x k(x)–17 –11–13 –5–9 10–5 –14If k(x) is continuous on [–17,–5], can you conclude that the equation k(x)=–4 has a solution?
Solution
Yes, we can conclude that the equation k(x) = -4 has a solution on the interval [-17, -5]. This is due to the Intermediate Value Theorem, which states that if a function is continuous on a closed interval [a, b], then it takes on every value between f(a) and f(b) at some point within the interval.
Looking at the table, we can see that k(x) takes on values both above and below -4 in the interval [-17, -5]. Specifically, at x = -13, k(x) = -5, and at x = -9, k(x) = 10. Since -4 is between -5 and 10, and k(x) is continuous, we can conclude that k(x) = -4 for some x in the interval [-13, -9] by the Intermediate Value Theorem.
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