At the point $x=-\pi$
, the function $c(x)=\cos 2 x+0.4 \cos 5 x$
has
A)A local maximum
B)A local minimum
C)Neither

Question

At the point $x=-\pi$
 , the function $c(x)=\cos 2 x+0.4 \cos 5 x$
 has
A)A local maximum
B)A local minimum
C)Neither

Quizplus · Accepted Answer

To determine if $c(x) = \cos 2x + 0.4 \cos 5x$ has a local maximum, minimum, or neither at $x = -\pi$, we can examine the first and second derivatives of $c(x)$.First derivative $c'(x) = -2\sin 2x - 2\sin 5x$.Second derivative $c''(x) = -4\cos 2x - 20\cos 5x$.At $x = -\pi$,$c'(-\pi) = -2\sin(-2\pi) - 2\sin(-5\pi) = 0$, indicating a potential maximum or minimum.$c''(-\pi) = -4\cos(-2\pi) - 20\cos(-5\pi) = -4\cos(2\pi) - 20\cos(5\pi) = -4(1) - 20(1) = -24$, which is less than 0, indicating a concave down curve at $x = -\pi$.Since the second derivative is negative, $c(x)$ has a local maximum at $x = -\pi$.

At the Point x=−πx=-\pix=−π , the Function c(x)=cos⁡2x+0.4cos⁡5xc(x)=\cos 2 x+0.4 \cos 5 xc(x)=cos2x+0.4cos5x

At the Point $x=-\pi$ , the Function $c(x)=\cos 2 x+0.4 \cos 5 x$