If f(x) = sin ( cos ⁻¹( (1 - 2²ˣ)/(1 + 2²ˣ) ) ) and its first derivative with respect to x is - b/alog ₑ2 when x = 1, where a and b are integers, then the minimum value of | a² - b² | is ____________ .

Solution

Maths sir · 4 steps

Turn sin(cos⁻¹ u) into √(1-u²) first, then the derivative is a plain quotient rule.

1. Simplify first. Let u = (1-4ˣ)/(1+4ˣ), using 2²ˣ = 4ˣ. Since sin(cos⁻¹ u) = √(1-u²), which is never negative, we get 1-u² = ((1+4ˣ)²-(1-4ˣ)²)/(1+4ˣ)² = (4· 4ˣ)/(1+4ˣ)².
2. Clean function. Taking the square root gives f(x) = (2· 2ˣ)/(1+4ˣ) = 2ˣ⁺¹/(1+4ˣ). Now differentiate with the quotient rule, using d/dx2ˣ⁺¹ = 2ˣ⁺¹ln 2 and d/dx4ˣ = 2· 4ˣ ln 2.

   f'(x) = (2ˣ⁺¹ln 2 (1+4ˣ) - 2ˣ⁺¹· 2· 4ˣ ln 2)/(1+4ˣ)² = 2ˣ⁺¹(1-4ˣ)/(1+4ˣ)²ln 2

3. Put x = 1. f'(1) = (4×(1-4))/(1+4)²ln 2 = -12/25 ln 2. So b/a = 12/25, and since 12 and 25 share no factor, a = 25k and b = 12k for some nonzero integer k.
4. So. |a² - b²| = k²(625 - 144) = 481k². This is smallest when k² = 1, so the minimum value is 481.

The common slip: 589. Writing √(4· 4ˣ) as 2ˣ instead of 2· 2ˣ gives f'(1) = -6/25 ln 2 and |a²-b²| = 589.

The idea to keep. When a trig function sits on an inverse trig function, convert it to an algebraic expression before you differentiate. And when only a ratio is fixed, scale it by k to find the minimum.

Answer from the NTA final answer key. Solution by PrepWiser.