Application of DerivativeHard
Question
If f : [1, 10] → [1, 10] is a non-decreasing function and g : [1, 10] → [1, 10] is a non-increasing function. Let h(x) = f(g(x)) with h(1) = 1, then h(2)
Options
A.lies in (1, 2)
B.is more than 2
C.is equal to 1
D.is not defined
Solution
x > 1 ε f(x) ≥ f(1)
x > 1 ⇒ g(x) ≤ g(1)
⇒ f(g(x)) ≤ f (g(1)) .... (i)
Range of h(x) is subset of [1, 10]
⇒ h(x) ≥ 1 .... (ii)
By (i), (ii) we have h(x) = 1 ⇒ h(2) = 1
x > 1 ⇒ g(x) ≤ g(1)
⇒ f(g(x)) ≤ f (g(1)) .... (i)
Range of h(x) is subset of [1, 10]
⇒ h(x) ≥ 1 .... (ii)
By (i), (ii) we have h(x) = 1 ⇒ h(2) = 1
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