$f(x)=\int_{0}^{x} 2\sin^4 t \cos^2 t dt$ is lipschitz continous right?

Question

$f(x)=\int_{0}^{x} 2\sin^4 t \cos^2 t dt$ is lipschitz continous right?

as $|f(x)-f(y)|\le2|x-y|$?

in general if $f$ is bounded then $F(x)=\int_{0}^{x} f(t)dt$ is lipschitz continous, am I right?

Answer 1

Just notice that

$$ f'(x)=2\sin(x)^4 \cos(x)^2 \implies |f'(x)|\leq 2. $$

Now, you can apply mean value theorem to prove the assertion.