Question

the graph of a function f is shown above. which of the following statements about f is false?
a f is continuous at x = a.
b f has a relative maximum at x = a.
c x = a is in the domain of f.

Explanation:

Step1: Analyze continuity

A function is continuous at a point if the limit as x approaches that point from the left equals the limit as x approaches from the right and equals the function - value at that point. From the graph, the function appears to be unbroken at \(x = a\), so it is continuous at \(x=a\).

Step2: Analyze relative - maximum

A relative maximum occurs at a point where the function changes from increasing to decreasing. At \(x = a\), the function is increasing (the slope of the tangent line is positive), so it does not have a relative maximum at \(x = a\).

Step3: Analyze the domain

Since there is a defined \(y\) - value for \(x=a\) on the graph, \(x = a\) is in the domain of \(f\).

Answer:

B. \(f\) has a relative maximum at \(x = a\)