the graph of F(x) = x0f(t) dt.(a) Explain why F(x) ≥ 0 on [0, 6].
Question
the graph of F(x) = x0f(t) dt.(a) Explain why F(x) ≥ 0 on [0, 6].
Solution
The function F(x) is defined as the integral from 0 to x of f(t) dt. This means that F(x) is the area under the curve of f(t) from 0 to x.
(a) F(x) ≥ 0 on [0, 6] because the integral of a function over an interval is the net area between the function and the x-axis over that interval. If the function f(t) is non-negative on the interval [0, 6], then the area under the curve of f(t) from 0 to x (for any x in [0, 6]) is non-negative. Therefore, F(x) is non-negative on [0, 6].
This is under the assumption that f(t) is non-negative on the interval [0, 6]. If f(t) is not non-negative on this interval, then we cannot say that F(x) is non-negative on [0, 6] without additional information.
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