Question

If a function $f$ is continuous for all $x$ and if $f$ has a relative maximum at $(-1,4)$ and a relative minimum at $(3,-2)$, then which of the following statements must be true?

• A. The graph of $f$ has a point of inflection somewhere between $x=-1$ and $x=3$.
• B. $f^{\prime }(-1)=0$.
• C. The graph of $f$ has a horizontal tangent line at $x=3$.
• D. The graph of $f$ intersect both co-ordinate axes.

Answer 1

Answer: (D)

Aaodsc Because $f$ is continuous for all $x$, the intermediate value theorem implies that the graph of $f$ must intersect the $x$-axis. The graph must also intersect the $y$-axis since $f$ is defined for all $x$, in particular, at $x=0$ As $f$ need not be differentiable.
Hence $A$, B and $C$ need not be correct.