Application of Linear Combination Theorem 
In $(\mathbb{R}^{2},\|\cdot\|_{p})$ with $1 \leqslant p < \infty$ and under the standard basis $\{e_{1},e_{2}\}$ find the largest possible $c_{p} > 0 $ satisfiying
  $$ \|a_{1}e_{1} + a_{2}e_{2} \|_{p} \geqslant c_{p}(|a_{1}| + |a_{2}|)$$
  for all $a_{1}, a_{2} \in \mathbb{R}$.

I have tried to work with the statment algebraically but with no luck as I do not think this is how to solve this. I am interested in any tips or hints or how to solve this. 

Consider the function of two variables
$$
f(x,y) = \frac{\sqrt[p]{x^p+y^p}}{x+y}
$$
for $x,y\geq 0$. What is the max/min of $f$ on the curve $x^p+y^p=1$? Why is it enough to study $f$ on that curve?
