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Most Important Selected Qs for JEE AdvancedMathematicsContinuity and Differentiability

Let f be a continuous and differentiable function in (x₁, x₂ ) and the following conditions hold for it. array l f(x) f^ (x) x 1-(f(x))^4 _ x x₁⁺ (f(x))^2=1 _ x x₂⁻ (f(x))^2= 1 2 array . If the minimum value of (x₁^2-x₂^2 ) is equal to k where k N , then find the value of k . [Note: f^ (x)= d f(x) d x ]

Correct answer

3

Step-by-step solution

aligned & _ x₁ ^ x₂ f(x) f^ (x) 1-(f(x))^4 d x _ x₁ ^ x₂ x d x & . 1 2 ⁻¹ f^2(x) ]_ x₁ ^ x₂ 1 2 (x₂^2-x₁^2 ) & ⁻¹ f^2 (x₂⁻ )- ⁻¹ f^2 (x₁⁺ ) (x₂^2-x₁^2 ) & 6 - 2 (x₂^2-x₁^2 ) & 3 (x₁^2-x₂^2 ) & x₁^2-x₂^2 3 & aligned Hence minimum value of x₁^2-x₂^2 is 3 k=3

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