State Intermediate -Value Theorem. Suppose f is continuous function on theinterval [a, b] where a < b.Show that there is at least one number c in theinterval [a, b] such that f (c) = c
Question
State Intermediate -Value Theorem. Suppose f is continuous function on theinterval [a, b] where a < b.Show that there is at least one number c in theinterval [a, b] such that f (c) = c
Solution 1
To prove the Intermediate Value Theorem, we need to show that for any continuous function f defined on the interval [a, b], where a < b, there exists at least one number c in the interval [a, b] such that f(c) = c.
Step 1: Define the function f(x) on the interval [a, b]. Step 2: Verify that f(x) is continuous on the interval [a, b]. This means that f(x) is defined and has no abrupt jumps or breaks within the interval. Step 3: Check if f(a) = a or f(b) = b. If either of these conditions is true, then we have found a number c in the interval [a, b] such that f(c) = c, and we are done. Step 4: If f(a) ≠ a and f(b) ≠ b, consider the function g(x) = f(x) - x. Step 5: Show that g(x) is continuous on the interval [a, b]. Since f(x) is continuous and x is continuous, their difference g(x) is also continuous. Step 6: Evaluate g(a) and g(b). If g(a) and g(b) have opposite signs, then by the Intermediate Value Theorem, there exists at least one number c in the interval (a, b) such that g(c) = 0. Step 7: Simplify g(c) = 0 to f(c) - c = 0, which implies f(c) = c. Step 8: If g(a) and g(b) have the same sign, we need to consider additional cases or apply further techniques to find a suitable c in the interval [a, b] such that f(c) = c.
By following these steps, we can prove the Intermediate Value Theorem and show the existence of at least one number c in the interval [a, b] such that f(c) = c.
Solution 2
To prove the Intermediate Value Theorem, we need to show that for any continuous function f defined on the interval [a, b], where a < b, there exists at least one number c in the interval [a, b] such that f(c) = c.
Step 1: Define the function f(x) on the interval [a, b]. Step 2: Verify that f(x) is continuous on the interval [a, b]. This means that f(x) is defined and has no abrupt jumps or breaks within the interval. Step 3: Check if f(a) = a or f(b) = b. If either of these conditions is true, then we have found a number c in the interval [a, b] such that f(c) = c, and we are done. Step 4: If f(a) ≠ a and f(b) ≠ b, consider the function g(x) = f(x) - x. Step 5: Show that g(x) is continuous on the interval [a, b]. Since f(x) is continuous and x is continuous, their difference g(x) is also continuous. Step 6: Evaluate g(a) and g(b). If g(a) and g(b) have opposite signs, then by the Intermediate Value Theorem, there exists at least one number c in the interval (a, b) such that g(c) = 0. Step 7: Simplify g(c) = 0 to f(c) - c = 0, which implies f(c) = c. Step 8: If g(a) and g(b) have the same sign, we need to consider additional cases or apply further techniques to find a suitable c in the interval [a, b].
By following these steps, we can prove the Intermediate Value Theorem and show that there is at least one number c in the interval [a, b] such that f(c) = c.
Solution 3
To prove the Intermediate Value Theorem, we need to show that for any continuous function f defined on the interval [a, b], where a < b, there exists at least one number c in the interval [a, b] such that f(c) = c.
Step 1: Define the function f(x) on the interval [a, b]. Step 2: Verify that f(x) is continuous on the interval [a, b]. This means that f(x) is defined and has no abrupt jumps or breaks within the interval. Step 3: Check if f(a) = a or f(b) = b. If either of these conditions is true, then we have found a number c in the interval [a, b] such that f(c) = c, and we are done. Step 4: If f(a) ≠ a and f(b) ≠ b, consider the function g(x) = f(x) - x. Step 5: Show that g(x) is continuous on the interval [a, b]. Since f(x) is continuous and x is continuous, their difference g(x) is also continuous. Step 6: Evaluate g(a) and g(b). If g(a) and g(b) have opposite signs, then by the Intermediate Value Theorem, there exists at least one number c in the interval (a, b) such that g(c) = 0. Step 7: Simplify g(c) = 0 to f(c) - c = 0, which implies f(c) = c. Step 8: If g(a) and g(b) have the same sign, we need to consider additional cases or apply further techniques to find a suitable c in the interval [a, b] such that f(c) = c. Step 9: Conclude that there is at least one number c in the interval [a, b] such that f(c) = c, based on the Intermediate Value Theorem.
By following these steps, we can prove the Intermediate Value Theorem for a continuous function f on the interval [a, b], where a < b.
Solution 4
To prove the Intermediate Value Theorem, we need to show that for any continuous function f defined on the interval [a, b], where a < b, there exists at least one number c in the interval [a, b] such that f(c) = c.
Step 1: Define the function f(x) on the interval [a, b]. Step 2: Verify that f(x) is continuous on the interval [a, b]. This means that f(x) is defined and has no abrupt jumps or breaks within the interval. Step 3: Check if f(a) = a or f(b) = b. If either of these conditions is true, then we have found a number c in the interval [a, b] such that f(c) = c, and we are done. Step 4: If f(a) ≠ a and f(b) ≠ b, consider the function g(x) = f(x) - x. Step 5: Show that g(x) is continuous on the interval [a, b]. Since f(x) is continuous and x is continuous, their difference g(x) is also continuous. Step 6: Evaluate g(a) and g(b). If g(a) and g(b) have opposite signs, then by the Intermediate Value Theorem, there exists at least one number c in the interval (a, b) such that g(c) = 0. Step 7: Simplify g(c) = 0 to f(c) - c = 0, which implies f(c) = c. Step 8: If g(a) and g(b) have the same sign, we need to consider additional cases or apply further techniques to find a suitable c in the interval [a, b] such that f(c) = c. Step 9: Conclude that there is at least one number c in the interval [a, b] such that f(c) = c, based on the Intermediate Value Theorem and the steps outlined above.
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