A function f(x) is continuous for all x and has a local minimum at (1,8). Which must be true?
A. f'(1)=0
B. f' exists at x=1
C. The graph is concave up at x=1
D. f'(x) is less than 0 if x is less than 1, f'(x) is greater than 0 if x is greater than 1
E. f'(x) is greater than 0 if x is less than 1, f'(x) is less than 0 if x is greater than 1
To find the correct answer, we need to understand the properties of a local minimum for a continuous function.
A local minimum occurs at a point (a, b) on a graph when the function is at its lowest value at that specific point. In other words, the y-value b is lower than or equal to all other y-values on either side of the point (a, b).
Let's analyze each option to determine which must be true:
A. f'(1) = 0: This statement refers to the derivative of the function at x=1. A local minimum occurs when the derivative changes from negative to positive at that specific point. So, f'(1) ≠ 0. Therefore, option A is not true.
B. f' exists at x=1: To have a local minimum at a point, f'(x) must exist and be finite on both sides of that point. Since the function has a local minimum at (1, 8), it is implied that f' exists at x=1. Therefore, option B is true.
C. The graph is concave up at x=1: The graph of a function is said to be concave up at a point when the function is increasing and the second derivative is positive at that point. However, this information is not provided in the question. Therefore, we cannot determine the concavity of the graph at x=1. Option C could be true or false.
D. f'(x) is less than 0 if x is less than 1, f'(x) is greater than 0 if x is greater than 1: A local minimum occurs when the derivative changes from negative to positive. This means that f'(x) is negative when x is less than 1 and f'(x) is positive when x is greater than 1. Therefore, option D is true.
E. f'(x) is greater than 0 if x is less than 1, f'(x) is less than 0 if x is greater than 1: This statement is the opposite of what we concluded in option D. Therefore, option E is not true.
In summary, the correct answer is B. f' exists at x=1.