Question

The graph of the function $\cos x\cos x(x+2)-{\cos }^2(x+1)$ is

• A. A straight line through $(0,-{\sin }^{2}1)$ with slope $2$.
• B. A straight line through $(0,0)$
• C. A parabola with vertex $(1,-{\sin }^{2}1)$
• D. A straight line through $(\frac{\pi }{2},-{\sin }^{2}1)$ and parallel to the $x-axis$.

Answer 1

The correct option is D A straight line through $(\frac{\pi }{2},-{\sin }^{2}1)$ and parallel to the $x-axis$.

$\begin{array}{cc}y& =\cos x\cos (x+2)-{\cos }^2(x+1)\\ & =\cos (x+1-1)\cos (x+1+1)-{\cos }^2(x+1)\\ & ={\cos }^2(x+1)-{\mathrm{lin}}^{2}1-{\cos }^2(x+1)\{\because \cos (A+B\left)\cos \right(A-\beta {)}^2\\ & =-{\sin }^{2}1.\end{array}$

for $y=-{\mathrm{lin}}^{2}1,x=\frac{\pi }{2}$

$f^{\prime }\left(x\right)=-\sin x\cos (x+2)-\cos x\sin (x+2)+2\cos (x+1)xsin(x+1)$

$=-\mathrm{lin}(2x+2)+\sin (2x+2)$

$=0$ i.c. parallel to $x$ -asis.

$\therefore$ option $D$ is corret.