$H^1$ norm estimation of an affine function

Question

Let $v(x)=\alpha +(\beta-\alpha)x$ a function in $H^1(\Omega)$ with $\Omega=[0,1]$ and $\alpha$ and $\beta$ are constants. How do we prove that there exist an constant $M >0$ such that $$||v||_{H^1} \leq M (|\alpha| + |\beta|)?$$ Thanks for the help.

Answer 1

Since $v(x)^2\leqslant 2\alpha^2+2(\beta-\alpha)^2x^2$, we have after an integration that $\lVert v\rVert_2\leqslant M_1\sqrt{\alpha^2+(\beta-\alpha)^2}$ where $M_1$ is a constant. This can be made small than $2M_1(|\alpha|+|\beta|)$ by the inequality $\sqrt{a+b}\leqslant \sqrt a+\sqrt b$ for non-negative $a$ and $b$.

Since $v'(x)=\beta-\alpha$, we have $\lVert v'\rVert_{L^2}\leqslant |\beta|+|\alpha|$.