Let ƒ(x) = 3x - x³ . Show that the equation ƒ(𝓍) = -4 has a solution in the interval [2,3] and use Newton’s method to find it.

Verified step by step guidance

1. First, verify that the function ƒ(x) = 3x - x³ has a solution in the interval [2,3] by applying the Intermediate Value Theorem. Calculate ƒ(2) and ƒ(3) to check if there is a sign change.
2. Calculate ƒ(2): ƒ(2) = 3(2) - (2)³ = 6 - 8 = -2. Calculate ƒ(3): ƒ(3) = 3(3) - (3)³ = 9 - 27 = -18. Since ƒ(2) = -2 and ƒ(3) = -18, there is no sign change, so check the values of ƒ(x) at the endpoints to ensure a solution exists.
3. Since the function is continuous and ƒ(2) = -2 and ƒ(3) = -18, check if the value -4 lies between these two values. Since -4 is between -2 and -18, by the Intermediate Value Theorem, there is at least one solution in the interval [2,3].
4. To apply Newton's method, start with an initial guess x₀ within the interval [2,3]. A reasonable choice is x₀ = 2.5. Newton's method formula is xₙ₊₁ = xₙ - ƒ(xₙ)/ƒ'(xₙ).
5. Find the derivative ƒ'(x) = 3 - 3x². Use this derivative in the Newton's method formula: xₙ₊₁ = xₙ - (3xₙ - xₙ³ + 4)/(3 - 3xₙ²). Substitute x₀ = 2.5 into this formula to find the next approximation x₁.

Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function takes on two values at two points, it must take on every value between those two points at least once. In this case, we can evaluate ƒ(2) and ƒ(3) to show that the function changes sign over the interval [2,3], indicating that there is at least one solution to the equation ƒ(𝓍) = -4 within that interval.

Newton's Method

Newton's Method is an iterative numerical technique used to find approximate solutions to equations. It starts with an initial guess and refines it using the formula x_{n+1} = x_n - ƒ(x_n)/ƒ'(x_n), where ƒ' is the derivative of ƒ. This method is particularly effective for finding roots of functions when the derivative is known and can lead to rapid convergence to the actual solution.

Derivatives

The derivative of a function measures how the function's output changes as its input changes. It is a fundamental concept in calculus that provides information about the function's slope and behavior. For Newton's Method, calculating the derivative of ƒ(x) is essential, as it is used to determine the slope at the current approximation, guiding the next guess towards the root.