Question

Consider the equation xe^x = cos x

(a) Apply the intermediate value theorem to show that the function has a root in the interval

[0, 1].

Answer 1

Function $f(x)=xe^x-\cos x$ is continuous on $[0, 1].$

$$f(0)=0(e^0)-\cos(0)=-1<0$$

$$f(1)=1(e^1)-\cos(1)=e-\cos (1)>0,$$

since $e>1, \cos(1)<1.$

Then by the intermediate value theorem there exists the number $c$ in $(0, 1)$ such that $f(c)=0.$

Therefore the function $f(x)=xe^x-\cos x$ has a root in the interval $[0, 1].$