Question

Verify Rolle's theorem for the function,f defined by

f(x)= x(x-2)e^-x,on the interval [0,2].

Answer 1

We have,

$f(x)= x(x-2)e^{-x}$

Therefore,

$f'(x) = (2x-2)e^{-x}-(x^2-2x)e^{-x}$

$f'(x)$ exists for every value of $x$ in the interval $[0,2].$

Hence, $f(x)$ is differentiable and hence, continuous in the interval $[0,2].$

Now,

$f(0) = -2$

$f(2) = 2e^{-2}$

$f(0) \ne f(2).$

Therefore Rolle's Theorem is not valid in this problem.