I could not prove this.
Opalq, thank you for your response. That is a very good proof. But I need to use the hint given in the question. That is I need to show the violation of
$\displaystyle \|cx\|_p=|c| \|x\|_p$ where c is a scalar, 0 < c < 1
I think you are misunderstanding the hint. You will never find a function x such that $\displaystyle \|cx\|_p \ne |c| \|x\|_p$, because $\displaystyle \|cx\|_p = \left[\int_0^1|cx(t)|^pdt\right]^{1/p} = \left[|c|^p\int_0^1|x(t)|^pdt\right]^{1/p} = |c|\|x\|_p$.
In fact, I used the hint in my example, in the inequality $\displaystyle (1/2)^{1/p}<1/2$, which is equivalent to $\displaystyle 1/2<(1/2)^p$.