Let be a function satisfying . Which of the following statements must be true?
B) for all
C) The tangent line to the graph of at every point has a positive slope.
D) is not differentiable at
E) None of the above.

Question

Let  be a function satisfying  . Which of the following statements must be true?
B) for all 
C) The tangent line to the graph of at every point has a positive slope. 
D) is not differentiable at 
E) None of the above.

Quizplus · Accepted Answer

Since the function satisfies the condition 11eaa655_3b45_2776_a696_216c23381c27_TB5596_11, it must be continuous at every point. If the tangent line to the graph of the function at a point has a positive slope, it means that the function is increasing at that point. Since the function is continuous, if it is increasing at one point, it must be increasing in a small interval around that point. Therefore, the statement C must be true. The other statements do not follow from the given information.

Let Be a Function Satisfying