Show that the equation x^3-15x+c=0 has at most one root in the interval [-2,2].
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To show that the equation x^3 - 15x + c = 0 has at most one root in the interval [-2,2], we can use the concept of the Intermediate Value Theorem.
The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes on two different values at the endpoints, then it must also take on every value between those two values at some point within the interval.
Let's consider the function f(x) = x^3 - 15x + c for the given interval [-2,2].
1. Evaluate the function at the endpoints:
f(-2) = (-2)^3 - 15(-2) + c = 8 + 30 + c = 38 + c
f(2) = (2)^3 - 15(2) + c = 8 - 30 + c = -22 + c
2. We need to determine whether f(-2) and f(2) have different signs or not. If they have different signs, then the function must have at least one root within the interval [-2,2].
3. If f(-2) and f(2) have the same sign, then we can conclude that there is no root or at most one root in the interval [-2,2]. This is because the function is continuous, and according to the Intermediate Value Theorem, it can't change sign without crossing the x-axis.
Let's consider the two cases:
Case 1: f(-2) and f(2) have different signs.
If f(-2) > 0 and f(2) < 0 (or vice versa), then there must be at least one root in the interval [-2,2]. In this case, the equation x^3 - 15x + c = 0 has at least one root in the interval, and it might have more.
Case 2: f(-2) and f(2) have the same sign.
If f(-2) and f(2) are both positive or both negative, then there cannot be any root or at most one root in the interval [-2,2]. In this case, the equation x^3 - 15x + c = 0 has at most one root in the interval.
Therefore, by analyzing the signs of f(-2) and f(2), we can determine that the equation x^3 - 15x + c = 0 has at most one root in the interval [-2,2].
To show that the equation x^3 - 15x + c = 0 has at most one root in the interval [-2, 2], we can make use of the Intermediate Value Theorem and the fact that a cubic function can only change sign at points where it crosses the x-axis.
Here's how we can proceed:
Step 1: Find the values of f(-2) and f(2).
Substitute x = -2 and x = 2 into the equation x^3 - 15x + c = 0 and determine the values of f(-2) and f(2).
For x = -2, we have: f(-2) = (-2)^3 - 15(-2) + c = -8 + 30 + c = 22 + c.
Similarly, for x = 2, we have: f(2) = 2^3 - 15(2) + c = 8 - 30 + c = -22 + c.
Step 2: Consider the cases where f(-2) and f(2) have the same sign or different signs.
Case 1: If f(-2) and f(2) have the same sign.
If both f(-2) and f(2) are positive or both are negative, then the function f(x) does not cross the x-axis in the interval [-2, 2]. In this case, there is no root in the interval [-2, 2].
Case 2: If f(-2) and f(2) have different signs.
If f(-2) and f(2) have different signs, then the function f(x) must cross the x-axis at least once in the interval [-2, 2]. In this case, there is at least one root in the interval [-2, 2].
Step 3: Final conclusion.
Based on the results from Case 1 and Case 2, we can conclude that the equation x^3 - 15x + c = 0 has at most one root in the interval [-2, 2].
Please note that the value of constant 'c' does not affect the number and location of the roots within the given interval, as it acts as a vertical shift of the graph rather than changing its shape.