For every pair of continuous functions f, g:[0, 1] → R such that
max {f(x) : x ∈ [0,1]} = max {g(x) : x ∈ [0, 1]},the correct statement(s) is (are) :
Text Solution
Verified by ExpertsA
(a,d)
Sol. Consider
h(x) = f(x) – g(x) Assume a < b
h(a) = λ – g(a) > 0
h(b) = f(b) – λ < 0
else if a > b h(a) < 0 and h(b) > 0.
By intermediate value theorem ⇒ h(c) = 0 .....
(f(c)) 2 + 3f(c) = (g(c)) 2 + 3g(c)
(f(c) – g(c)) (f(c) + g(c) + 3) = 0
So there exist a 'c' : f(c) – g(c)
from .
Hence A is correct.
Similarly (f(c)) 2 = (g(c)) 2
(f(c) – g(c)) (f(c) + g(c)) = 0
⇒ is correct.
B & C are wrong as by counter eg
If f(x) = g(x) = λ ≠ 0, then
B → λ 2 + λ = λ 2 + 3 λ is not possible.
C → λ 2 + 3 λ = λ 2 + λ is not possible.
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