Increasing and decreasing function in certain interval

[harimakenji]

Homework Statement

P(x) is an increasing function and Q(x) is a decreasing function in interval a ≤x≤b, x is positive. P(x) and Q(x) are located in 1^st quadrant. Another function γ(x) satisfies m≤γ(x)≤M.
Find the value of m and M if:
a. γ(x) = P(x) - Q(x)
b. γ(x) = P(x) . Q(x)
c. γ(x) = [P(x)]² – [Q(x)]²
d. γ(x) = 1/P(X) +Q(x)
e. γ(x)= (P(x))/(Q(x))-(Q(x))/(P(x))

Homework Equations

Maybe differentiation

The Attempt at a Solution

P(x) is increasing function = P'(x) is positive and P(a) < P(b)
Q(x) is decreasing function = Q'(x) is negative and Q(a) > Q(b)

a. γ'(x) = P'(x) - Q'(x). Since Q'(x) is negative, γ'(x) will be positive so γ(x) is increasing function.
m = P(a) - Q(b) and M = P(b) - Q(b). Is this correct?

b. γ'(x) = P'(x).Q(x) + P(x).Q'(x). I can not determine whether γ'(x) is positive or negative so I don't understand how to find m and M

c. γ'(x) = 2 P(X) P'(x) - 2 Q(x) Q'(x). The value of γ'(x) is positive so m = P²(a) - Q²(a) and M = P²(b) - Q²(b). Is this correct?

d. γ'(x) = -P'(x) / P²(x) + Q'(x). The value of γ'(x) is negative so m = 1/P(b) + Q(b) and M = 1/P(a) + Q(a). Is this correct?

e. γ'(x) = \frac{P&#039;(x).Q(x)-P(x).Q&#039;(x)}{Q^{2}(x)} - \frac{Q&#039;(x).P(x)-P&#039;(x).Q(x)}{P^{2}}. The value of γ'(x) is positive so m = (P(a))/(Q(a))-(Q(a))/(P(a)) and M = (P(b))/(Q(b))-(Q(b))/(P(b)). Is this correct?

Please help me to verify my answer and guide me how to solve (b). Thank you very much.

 

[tiny-tim]

hi harimakenji!

your a c d and e look fine

i don't see how you can solve b … it could be anything (wcihi you can see more clearly by taking logs: A = log P, B = logQ, so A is any increasing function, B is any decreasing function, and you want max and min of A + B)

 

[harimakenji]

hi harimakenji!

your a c d and e look fine

i don't see how you can solve b … it could be anything (wcihi you can see more clearly by taking logs: A = log P, B = logQ, so A is any increasing function, B is any decreasing function, and you want max and min of A + B)

Hi tiny-tim. Thank you for the response

I don't really get what you are trying to tell me by using the logs. And one more thing, actually the real question doesn't mention anything about P(x) and Q(x) being in the first quadrant. I just added it myself.

If there is no information about the first quadrant, I think we can only answer questions (a) and (d). Do you have the same opinion?

Thank you very much

 

[tiny-tim]

hi harimakenji!

If there is no information about the first quadrant, I think we can only answer questions (a) and (d). Do you have the same opinion?

yes

(and forget the logs, it doesn't matter)

 

[harimakenji]

hi harimakenji!


yes

(and forget the logs, it doesn't matter)

Ok, I will forget about the logs for now.

Thank you very much for your help