KVPY 2019 SB SX Exam Previous Papers

Question : 87 of 120

Marks: +1, -0

Let f : (–1, 1) → R be a differentiable function satisfying

(\f'(x))^4 = 16(\f(x))^2

for all x ∈ (–1, 1), f(0) = 0. The number of such functions is

Solution:

(\f '(x))^2 = ± 4\f(x)

Let f(x) is non negative in its domain
⇒

(\f '(x))^2 = = 4\f(x)

⇒

\f'(x) = ± 2√{\f(x)}

⇒

∫{\f '(x)}/√{\f(x)}dx=∫± 2dx\;\;\;

(f(x) ≠ 0)
⇒

2√{\f(x)}±2x \;\;\;

(f(0) = 0)
⇒

√{\f(x)}±x⇒\f(x) = x^2

so f(x) can be 0 or

x^2

proceeding in same way, we can also
have

\f(x) = -x^ 2

Also

\f(x) =\{\table x^2 {, },x< 0;-x^2 {, },x≥0

\f(x) =\{\table -x^2 {, },x< 0; x^2 {, },x≥0

etc

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