TRUE or FALSE: Suppose a function, f(x), is continuous near c. If f'(. c) = 0 and f”(. c) < 0, then f(x) has a local minimum at x=c.
True
True.
If a function, f(x), is continuous near c and f'(c) = 0, it means that the slope of the function at x = c is flat or has a horizontal tangent. This suggests that the function changes direction at x = c, which could indicate a local extremum.
To determine whether it is a local minimum or maximum, we look at the second derivative, f”(c). If f”(c) < 0, it means the rate of change of the slope is negative at x = c. According to the Second Derivative Test, if f"(c) < 0, then f(x) has a local maximum at x = c. However, if f"(c) > 0, it has a local minimum. Since the statement specifies that f”(c) < 0, it implies that f(x) has a local minimum at x = c. Therefore, the statement is true.
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