$f(x) = (\cos x - \sin x) (17 \cos x -7 \sin x) $ $f(x) = (\cos x - \sin x) (17 \cos x -7 \sin x)$
Determine the greatest and least values of $\frac{39}{f(x)+14}$ and state a value of x at which greatest values occurs. 
Do I just use a graphing calculator for this? Is there a way I could do this without a graphing calculator?

you can do this without calculus. here is a way.  i will use $t$ for $x.$ we have 
$$\begin{align}y &=  14 + (\cos t - \sin t)(17 \cos t - 7 \sin t) \\&= 14 + 17 \cos^2 t-24 \sin t \cos t+7 \sin ^2 t\\
&=14 + \frac{17}{2}(1 + \cos 2t)-12 \sin 2t+\frac 72(1-\cos 2t)\\
&=26+5\cos 2t-12\sin 2t\\
&=26 + 13\cos(2t+\phi), \text{ where} \cos \phi = \frac 5{13}, \sin \phi = \frac{12}{13} \end{align}$$  so the maximum value of $y$  is $39$ and minimum value of $y$ is $13.$ 
therefore  $$1 \le \frac {39}{ 14 + (\cos t - \sin t)(17 \cos t - 7 \sin t)} \le 3.$$
