Answer: the curve defined by the piecewise function f(x) has a discontinuity at x = 0, and there is no tangent line at this point due to the mismatch in slopes from the left and right sides of x = 0.
Explanation:
To determine if the curve has a tangent at x = 0, we need to check if the function is continuous at that point and if the derivatives from both sides match.
Let's analyze the function f(x):
For x < 0, the function is given as f(x) = 2 - 2x - x^2.
For x ≥ 0, the function is given as f(x) = 2x + 2.
First, we'll calculate the derivatives of the two parts of the function:
For x < 0:
f'(x) = d/dx (2 - 2x - x^2) = -2 - 2x
For x ≥ 0:
f'(x) = d/dx (2x + 2) = 2
Now, let's check if the function is continuous at x = 0:
For x < 0:
As x approaches 0 from the left side (x < 0), we have:
lim (x -> 0-) f(x) = lim (x -> 0-) (2 - 2x - x^2) = 2 - 0 - 0 = 2
For x ≥ 0:
As x approaches 0 from the right side (x ≥ 0), we have:
lim (x -> 0+) f(x) = lim (x -> 0+) (2x + 2) = 0 + 2 = 2
Since the left-hand and right-hand limits both approach the same value (2) as x approaches 0, the function is continuous at x = 0.
Now, let's check if the derivatives match at x = 0:
For x < 0:
f'(0-) = -2 - 2(0) = -2
For x ≥ 0:
f'(0+) = 2
The derivatives do not match at x = 0. Therefore, the function does not have a tangent at x = 0 because the slopes from the left and right sides of x = 0 are different.
In summary, the curve defined by the piecewise function f(x) has a discontinuity at x = 0, and there is no tangent line at this point due to the mismatch in slopes from the left and right sides of x = 0.
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