Show that the sum of two concave functions is concave. is the product of two concave functions also concave?

Answer with explanation:

Let us assume that the 2 functions are:

1) f(x)

2) g(x)

Now by definition of concave function we have the first derivative of the function should be strictly decreasing thus for the above 2 function we conclude that

Now the sum of the 2 functions is shown below

Diffrentiating both sides with respect to 'x' we get

Since each term in the right of the above equation is negative thus we conclude that their sum is also negative thus

Thus the sum of the 2 functions is also a concave function.

Part 2)

The product of the 2 functions is shown below

Diffrentiating both sides with respect to 'x' we get

Now we can see the sign of the terms on the right hand side depend on the signs of the function's themselves hence we remain inconclusive about the sign of the product as a whole. Thus the product can be concave or convex.

answer: all of the odd numbers were opened

the first person was the one that was opening all of the lockers. since one is odd, this causes all of the odd number lockers to be opened in the end.

i can't solve unless i know what the line is

step-by-step explanation:

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