Find the points of inflection and discuss the concavity of the function $f ( x ) = - \sin x + \cos x \text { on the interval } ( 0,2 \pi )$ .
A) no inflection points. concave up on $( 0,2 \pi )$
B) concave upward on $\left( 0 , \frac { 1 } { 2 } \pi \right)$; concave downward on $\left( \frac { 5 } { 2 } \pi , 2 \pi \right)$; inflection point at $\left( 0 , \frac { 1 } { 2 } \pi \right)$
C) no inflection points. concave down on $( 0,2 \pi )$
D) concave downward on $\left( 0 , \frac { 1 } { 2 } \pi \right)$; concave upward on $\left( \frac { 5 } { 2 } \pi , 2 \pi \right)$; inflection point at $\left( 0 , \frac { 1 } { 2 } \pi \right)$
E) none of the above

Question

Find the points of inflection and discuss the concavity of the function $f ( x ) = - \sin x + \cos x \text { on the interval } ( 0,2 \pi )$ . 
A) no inflection points. concave up on $( 0,2 \pi )$
B) concave upward on $\left( 0 , \frac { 1 } { 2 } \pi \right)$; concave downward on $\left( \frac { 5 } { 2 } \pi , 2 \pi \right)$; inflection point at $\left( 0 , \frac { 1 } { 2 } \pi \right)$
C) no inflection points. concave down on $( 0,2 \pi )$
D) concave downward on $\left( 0 , \frac { 1 } { 2 } \pi \right)$; concave upward on $\left( \frac { 5 } { 2 } \pi , 2 \pi \right)$; inflection point at $\left( 0 , \frac { 1 } { 2 } \pi \right)$
E) none of the above

Quizplus · Accepted Answer

To find the points of inflection and discuss the concavity of $f(x) = -\sin x + \cos x$, we first find the first and second derivatives of $f(x)$. The first derivative $f'(x) = -\cos x - \sin x$, and the second derivative $f''(x) = \sin x - \cos x$. The points of inflection occur where $f''(x) = 0$, which is when $\sin x = \cos x$. This occurs at $x = \frac{\pi}{4}$ and $x = \frac{5\pi}{4}$ within the interval $(0, 2\pi)$. The concavity changes at these points, so none of the given options correctly describe the concavity changes and inflection points for the entire interval $(0, 2\pi)$.

Find the Points of Inflection and Discuss the Concavity of the Function