Question
Mathematics
The figure shows the Graph of $f$f
inflection sfows the graph of a function $f$f. The zero and extrema for $f$f are labeled, and the point of Of the of the graph of $f$f is labeled. Let A, B, C, D. and $E$E represent the $x$x-coordinates at those points. Of the following, on which interval is $f$f decreasing and the graph of $f$f concave down?
(A) the interval from $A$A to $B$B
(B) the interval from $B$B to $C$C
(C) the interval from $C$C to $D$D
(D) the interval from $D$D to $E$E
(A) the interval from $A$A to $B$B.
From the graph, we can see that $f$f is decreasing on the interval from $A$A to $B$B, because the slope is negative in that region.
To determine whether $f$f is concave down on this interval, we can look at the shape of the graph. Since $f$f has an inflection point at $C$C, the concavity of the graph must change at that point. Specifically, $f$f is concave up on the interval from $A$A to $C$C, and concave down on the interval from $C$C to $E$E. Therefore, $f$f is concave down on the interval from $C$C to $D$D as well.
Since $f$f is decreasing and concave down on the interval from $A$A to $B$B, the correct answer is (A) the interval from $A$A to $B$B.