True or False
If f'(c)=0=f''(c)is less than 0, the f(c) is a local maximum. Justify your answer.
To determine whether the given statement is true or false, we need to analyze the properties of functions based on the given conditions.
In this case, we are given that f'(c) = 0 and f''(c) < 0. This means that the first derivative of the function is zero at c, indicating a critical point, and the second derivative at c is negative.
To understand the significance of these conditions, we need to recall the Second Derivative Test. According to this test:
1. If f'(c) = 0 and f''(c) > 0, then f(c) is a local minimum.
2. If f'(c) = 0 and f''(c) < 0, then f(c) is a local maximum.
3. If f'(c) = 0 and f''(c) = 0, the test is inconclusive.
Since f'(c) = 0 and f''(c) < 0, the conditions satisfy statement 2 of the Second Derivative Test. Therefore, f(c) is indeed a local maximum.
So, the answer to the question is True.