Increasing and Decreasing Function

P(*x*) is an increasing function and Q(*x*) is a decreasing function in interval $\displaystyle a\leq x\leq b$ , *x* is positive. Another function $\displaystyle \gamma (x)$ satisfies $\displaystyle m\leq \gamma (x)\leq M$. Find the value of m and M if:

a.$\displaystyle \gamma (x)$= P(*x*) . Q(x)

b.$\displaystyle \gamma (x)= [P(x)]^2 - [Q(x)]^2$

c. $\displaystyle \gamma (x) =\frac{1}{P(x)}+Q(x)$

d. $\displaystyle \gamma (x) = \frac{P(x)}{Q(x)}-\frac{Q(x)}{P(x)}$

Attempt :

$\displaystyle P(a)>P(b)$ and $\displaystyle Q(a)<Q(b)$

I can't solve this problem because I think we need to know that kind of functions P(x) and Q(x) are.

My thoughts :

1. If P(x) is exponential function and Q(x) is linear function, the result will be different compared to P(x) is linear and Q(x) is exponential. It also will be different if both P(x) and Q(x) are linear.

2. Even for both P(x) and Q(x) are linear, the answer still can't be determined. Assume P(a) = 2, P(b) = 4, Q(a) = 4, and Q(b) = 2. Then for $\displaystyle \gamma (x) = P(x)*Q(x)$ , P(a)*Q(a) = 8 and P(b)*Q(b)=8.

The result will be different if we assume P(a) = 0, P(b) = 1000, Q(a) = 4, Q(b) = 2.

Or there are flaws in my attempts?

Thx