An equation of the tangent line to the graph of at both and is . Which of the following statements must be true?
A) The function is the line .
D) The function is not differentiable at and at .
E) None of the above.

Question

An equation of the tangent line to the graph of  at both  and  is  . Which of the following statements must be true?
A) The function is the line . 
D) The function is not differentiable at and at . 
E) None of the above.

Quizplus · Accepted Answer

Since the line equation is given as the equation of the tangent line at two points on the graph of the function, it must be the derivative of the function at those points. Therefore, the function must be differentiable at those points, and option D cannot be true. The line equation matches the form of the derivative of the function 11eaa655_3b44_3cf9_a696_abe4e84538f4_TB5596_11 at those points, so option C is true. Option A is not necessarily true as the function could be another linear function that also passes through those two points. Option B is not necessarily true as we do not have enough information about the concavity of the function. Therefore, the correct answer is C.

An Equation of the Tangent Line to the Graph of