The function f is continuous on the closed interval [-5,5], and f(-2) = 6, f(1) = -3, and f(4) = 6. Which of the following statements must be true?
A. The equation f(x) = 0 has at least two solutions on the closed interval [-5,5].
B. The equation f(x) = 0 has exactly two solutions on the closed interval [-5,5]
C. The equation f'(x) = 0 has at least one solution on the closed interval [-5,5].
D. The equation f'(x) = 3 has at least one solution on the open interval (1,4).
E. The graph of f has at least one point of inflection on the closed interval [-5,5].
I can't choose between A and C. The graph of f(x) changes between positive and negative twice, so A might be the answer. But that also means it has a positive and negative slope, so C also seems correct.
A,C,D are true.
The Intermediate Value theorem says that there is a zero in (-2,1) and (1,4)
Rolle's Theorem (or MVT) says that f'(x)=0 somewhere in (-2,4)
Mean Value Theorem says D is true
To determine which statements are true, let's analyze the given information and the options provided.
Statement A states that the equation f(x) = 0 has at least two solutions on the closed interval [-5,5]. To determine if this is true, we need to examine the given function values. Since f(-2) = 6, f(1) = -3, and f(4) = 6, we can infer that the graph of f(x) intersects the x-axis at least twice within the interval [-5,5]. Therefore, Statement A is true.
Statement C states that the equation f'(x) = 0 has at least one solution on the closed interval [-5,5]. To determine if this is true, we need to analyze the derivative of the function f(x). However, the problem does not provide any information about the derivative f'(x) or its behavior. Therefore, we cannot determine whether Statement C is true based on the given information. It may be true depending on the specific properties of the function, but we cannot conclude it from the given information.
Therefore, the correct option is A. The statement "The equation f(x) = 0 has at least two solutions on the closed interval [-5,5]" must be true based on the given information.
To determine which statement must be true, let's analyze each option:
A. The equation f(x) = 0 has at least two solutions on the closed interval [-5,5].
This statement is not necessarily true. The fact that f(x) changes sign between -2, 1, and 4 does not guarantee that it crosses the x-axis (f(x) = 0) at least twice between -5 and 5. For example, the function could have only one solution that falls outside the interval.
C. The equation f'(x) = 0 has at least one solution on the closed interval [-5,5].
This statement is true based on the given information. If f(x) is continuous on the closed interval [-5,5], and it changes sign between -2, 1, and 4, it implies that the derivative, f'(x), must be zero at some point on the interval. This indicates that the function has a local extremum.
Therefore, the correct option is C. The equation f'(x) = 0 has at least one solution on the closed interval [-5,5].