Let f be a differentiable function defined on the closed interval [a,b] and let c be a point in the open interval (a,b) such that
I.f'(c)=0
II.f'(x)>0 when a≤x<c
III.f'(x)<0 when c<x<≤b
Which of the following statements must be true?
(A)f(c)=0
(B)f"(c)=0
(C)f(c) is an absolute maximum value of f on [a,b]
(D)f(c) is an absolute minimum value of f on [a,b]
(E)f(x) has a point of inflection at x=c
The correct answer is (C) f(c) is an absolute maximum value of f on [a,b].
Explanation:
Since f'(c) = 0, we know that the derivative of f at the point c is zero. This means that at c, the graph of f has a horizontal tangent line.
Now let's consider the intervals (a, c) and (c, b). According to statement II, f'(x) > 0 for x in the interval (a, c). This means that f is increasing on this interval.
According to statement III, f'(x) < 0 for x in the interval (c, b). This means that f is decreasing on this interval.
Combining these observations, we can conclude that f has a local maximum at c. Since c is in the closed interval [a, b], this local maximum is also an absolute maximum value of f on [a, b].
Therefore, (C) f(c) is an absolute maximum value of f on [a,b] is the correct statement.
To determine which statements must be true, let's analyze each option.
(A) f(c) = 0: This statement is not necessary. Just because the derivative, f'(c), is equal to 0 does not mean that f(c) must be equal to 0. The derivative indicates the slope or rate of change at a point, not the actual value of the function at that point.
(B) f"(c) = 0: This statement is also not necessarily true. The fact that f'(x) is positive on the interval [a, c] and negative on the interval [c, b] suggests a change in concavity at x = c. However, this does not guarantee that the second derivative, f''(x), is equal to 0 at x = c. The second derivative can be positive or negative, indicating upward or downward concavity, respectively.
(C) f(c) is an absolute maximum value of f on [a, b]: This statement is not necessarily true either. The fact that f'(x) is positive to the left of c and negative to the right of c implies that f(x) is increasing before c and decreasing after c, which suggests that f(c) may be a local maximum. However, this does not guarantee that f(c) is the absolute maximum value of f over the entire interval [a, b].
(D) f(c) is an absolute minimum value of f on [a, b]: This statement is also not necessarily true. Similar to statement (C), the behavior of f'(x) suggests that f(c) could be a local minimum, but it does not ensure that it is the absolute minimum value on the interval [a, b].
(E) f(x) has a point of inflection at x = c: This statement must be true. Given that f'(x) changes sign from positive to negative at x = c, this indicates a change in concavity at x = c, which implies that f(x) has a point of inflection at x = c.
Therefore, the only statement that must be true is (E) f(x) has a point of inflection at x = c.