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

Engineering

Mathematics

Maxima and Minima

Question

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

(f(c))² + f(c) = (g(c))² + 3g(c) for some c ∈ [0, 1].

(f(c))² = (g(c))² for some c ∈ [0, 1].

(f(c))² + 3f(c) = (g(c))² + g(c) for some c ∈ [0, 1].

(f(c))² + 3f(c) = (g(c))² + 3g(c) for some c ∈ [0, 1].

College PredictorLive

Know your College Admission Chances Based on your Rank/Percentile, Category and Home State.

Get your JEE Main Personalised Report with Top Predicted Colleges in JoSA

f (x₀) = g (x₀)

f (c₁) = g (c₂)

Aliter: Since f (x) and g (x) are 2 continuous function in [0, 1]

$∴ \exists α, β \in$ [0, 1]

such that f (a) = M and g (b) = M (where M is the maximum value)

Now consider a function h (x) = f (x) – g (x) in [α, β]

h (a) = M – g (a) $\geq$ 0

h (b) = f (b) – M $\leq$ 0

$∴$ h (a) · h (b) $\leq$ 0

⇒ Using I.V.T. $\exists c \in$ [0, 1] such that h (c) = 0

hence f (c) = g (c) ⇒ (A) & (D).