Final answer:
The function f(x) is differentiable at x = -2 because the derivative of f(x) exists at x = -2.
Step-by-step explanation:
The function f(x) is differentiable at x = -2 if the derivative of f(x) exists at x = -2. To check if f(x) is differentiable, we need to find the derivative of f(x) and evaluate it at x = -2.
To find the derivative, we can use the rules of differentiation. The derivative of 2x³ is 6x², the derivative of 3x² is 6x, and the derivative of 1 is 0. So, the derivative of f(x) = 2x³ - 3x² + 1 is f'(x) = 6x² - 6x.
To check if f(x) is differentiable at x = -2, we substitute x = -2 into f'(x) and see if it exists. Plugging in x = -2, we get f'(-2) = 6(-2)² - 6(-2) = 6(4) + 12 = 36 + 12 = 48.
The question concerns determining if the function f(x) is differentiable at x=−2. The function is given in piecewise form: when x ≤ 2, f(x) = 2x³−3x²+1, and when x > 2, f(x) = 3x−1. To check for differentiability at x=-2, we only need to consider the part of the function for x ≤ 2 since -2 falls in this interval. We need to find the derivative of this part of the function and check if it exists at x = -2.
To find the derivative, we use the power rule: f'(x) = d/dx of (2x³−3x²+1) = 6x²−6x. We then evaluate the derivative at x = -2: f'(-2) = 6(-2)²−6(-2) = 24 + 12 = 36. Since the derivative exists and is finite, f(x) is differentiable at x = -2.
Since the derivative of f(x) exists at x = -2 (f'(-2) = 48), we can conclude that the function f(x) is differentiable at x = -2.