Find Minum Value

Question

I have given this equation

Min(a-Q,b-W,c-E,d-R) =  a-Q

Now i have to find the answer of the following equation

Min(a-K* Q, b-K* W, c-K* E, d-K* R)  and k>=0

From my intuition i thinks it's always a-K*Q , But how to prove this ?

Answer 1

Your intuition is correct only for $k\geq1$. It fails when $k$ lies between $0$ and $1$.

For e.g. $a=3$, $b=1$, $Q=4$, $W=1$, $k=0.5$

Here $a-Q<b-W$ but $a-kQ>b-kW$.

Following is a proof for $k\geq1$,

$a-Q<b-W$

=> $a-b<Q-W$

=> $a-b<k(Q-W)$ for $k\geq1$

=> $a-kQ<b-kW$

Similarly for c-kE and d-kR