Question

The function f : x ı→ a + b cos x, is defined for 0 ≤ x ≤ 2π. Given that f(0) = 10 and that = 1 , find

(i) the values of a and b, (ii) the range of f, (iii) the exact value of

Answer 1

$f(x)=a+b\cos x\\ f(0)=a+b\cos 0=10\to max\\ a+b=10\\ f(\pi)=a+b\cos \pi=1\to min\\ a-b=1\\ \left\{\begin{matrix} a +b=10 \\ a-b=1 \end{matrix}\right.\\ i)\\ a=5.5\\ b=4.5\\ ii)\\ \cos x\in[-1,1]\\ 4.5\cos x\in[-4.5,4.5]\\ f(x)=5.5+4.5\cos x\in[1,10]\\ iii)\\ f(x)=5.5+4.5\cos x\\ f(\frac{\pi}{2})=5.5+4.5\cdot\cos\frac{\pi}{2}=5.5\\ f(\frac{\pi}{3})=5.5+4.5\cdot\cos\frac{\pi}{3}=7.75$