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If f(x) belongs to the interval [a,b], at which values of x does the curve y = f(x) have a tangent line parallel to the line y = mx + c?

User Jeffmayeur
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Final answer:

The values of x at which the curve y = f(x) has a tangent line parallel to the line y = mx + c are any points between 0 and 20 where the slope m equals to 0.

Step-by-step explanation:

To determine the values of x at which the curve y = f(x) has a tangent line parallel to the line y = mx + c, we need to understand that the tangent line to a curve at a given point will have the same slope as the curve at that point. For a line y = mx + c, the slope is represented by m. Since f(x) is said to form a horizontal line for 0 ≤ x ≤ 20, this implies that its slope is 0 within this interval. Thus, a tangent line to the curve f(x) would be parallel to any line with a slope of 0. In other words, for the curve y = f(x), a tangent line parallel to y = mx + c would exist at any point within the interval [0, 20] where m is equal to 0.

In this particular case, the function f(x) is already a horizontal line, so its tangent is the function itself. Therefore, anywhere between x = 0 and x = 20, the curve will have a tangent line parallel to any line where b = 0, such as y = c.

User Hoang Phan
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