Q:

Compare the functions shown below:f(x)cosine graph with points at 0, negative 1 and pi over 2, 1 and pi, 3 and 3 pi over 2, 1 and 2 pi, negative 1 g(x)x y−6 −11−5 −6−4 −3−3 −2−2 −3−1 −60 −11h(x) = 2 cos x + 1Which function has the greatest maximum y-value?a. f(x)b. g(x)c. g(x) and h(x)d. f(x) and h(x)

Accepted Solution

A:
We can find the local maxima and minima of any -continous- function by first finding where the slope is 0, as at this point maxima or minima exist.

Given an arbitrary function '$$f(x)$$' we find the point of slope 0 by taking its first derivative and equaling to 0 ('$$\frac{d}{dx}f(x)=0$$').

Lets, first, find the local extremes of the first function:
$$f(x)=cos(x)$$
$$\frac{d}{dx} f(x)=\frac{d}{dx}cos(x)=-sin(x)=0$$
$$x=sin^{-1} (0)=0$$

So our first function has a maxima at '$$x=0$$' or at '$$y=f(0)=cos(0)=1$$'.

Now we get the extremes for the second function:
$$h(x)=2cos(x)+1$$
$$\frac{d}{dx} h(x)=\frac{d}{dx}[2cos(x)+1]=-2sin(x)=0$$
$$x=0$$

So our second function has a maxima at '$$x=0$$' or at '$$y=h(0)=2cos(0)+1=3$$'.

Clearly, '$$h(0)=3\ \textgreater \ f(0)=1$$', this means the second function '$$h(x)$$' has the largest maxima -y value-.