Application of DerivativeHard
Question
Let f and g be two differentiable functions defined on an interval I such that f(x) ≥ 0 and g(x) ≤ 0 for all x ∈ I and f is strictly decreasing on I while g is strictly increasing on I then
Options
A.the product function fg is strictly increasing on I
B.the product function fg is strictly decreasing on I
C.fog(x) is monotonically increasing on I
D.fog (x) is monotonically decreasing on I
Solution
Let h(x) = f(x) g(x)
h′(x) = f′(x) g(x) + g′(x) f(x)
As f′(x) < 0, g(x) ≤ 0 ⇒ f′(x) g(x) ≥ 0
and g′(x) > 0, f(x) ≥ 0 ⇒ f(x) g′(x) ≥ 0
⇒ h′(x) ≥ 0
⇒ h(x) is increasing.
Let x1, x2 ∈ I
x1 < x2
g(x1) < g(x2)
f(g(x1)) > f(g(x2))
fog(x1) > fog(x2)
⇒ fog(x) is monotonically decreasing.
h′(x) = f′(x) g(x) + g′(x) f(x)
As f′(x) < 0, g(x) ≤ 0 ⇒ f′(x) g(x) ≥ 0
and g′(x) > 0, f(x) ≥ 0 ⇒ f(x) g′(x) ≥ 0
⇒ h′(x) ≥ 0
⇒ h(x) is increasing.
Let x1, x2 ∈ I
x1 < x2
g(x1) < g(x2)
f(g(x1)) > f(g(x2))
fog(x1) > fog(x2)
⇒ fog(x) is monotonically decreasing.
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