question 6 (1 point) suppose f(x) is continuous on 1,2, f(1)=5, and f(2)=9. by the intermediate value…

question 6 (1 point) suppose f(x) is continuous on 1,2, f(1)=5, and f(2)=9. by the intermediate value theorem, we can conclude (choose all that apply): there exists a c between 1 and 2 such that f(c)=0. there exists a c between 1 and 2 such that f(c)=10. there exists a c between 1 and 2 such that f(c)=8. f(1.5)=7 for any y between 5 and 9, there exists a c between 1 and 2 such that f(c)=y. view hint for question 6

question 6 (1 point) suppose f(x) is continuous on 1,2, f(1)=5, and f(2)=9. by the intermediate value theorem, we can conclude (choose all that apply): there exists a c between 1 and 2 such that f(c)=0. there exists a c between 1 and 2 such that f(c)=10. there exists a c between 1 and 2 such that f(c)=8. f(1.5)=7 for any y between 5 and 9, there exists a c between 1 and 2 such that f(c)=y. view hint for question 6

Answer

Explanation:

Step1: Recall Intermediate Value Theorem

If (y = f(x)) is continuous on ([a,b]), and (k) is a number between (f(a)) and (f(b)), then there exists at least one number (c\in(a,b)) such that (f(c)=k). Here (a = 1), (b = 2), (f(1)=5) and (f(2)=9).

Step2: Analyze each option

  • Option 1: Since (0) is not between (5) and (9), we cannot conclude there exists a (c\in(1,2)) such that (f(c)=0).
  • Option 2: Since (10) is not between (5) and (9), we cannot conclude there exists a (c\in(1,2)) such that (f(c)=10).
  • Option 3: Since (8) is between (5) and (9), by the Intermediate - Value Theorem, there exists a (c\in(1,2)) such that (f(c)=8).
  • Option 4: Just because (f(x)) is continuous on ([1,2]) and (f(1) = 5), (f(2)=9), we cannot say (f(1.5)=7). There is no information about the linearity or the exact form of (f(x)).
  • Option 5: By the Intermediate - Value Theorem, for any (y) between (5) and (9), there exists a (c\in(1,2)) such that (f(c)=y).

Answer:

There exists a (c) between (1) and (2) such that (f(c)=8); For any (y) between (5) and (9), there exists a (c) between (1) and (2) such that (f(c)=y)