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Differentiation under Integral Sign

20. If f(x) i...

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20. If f(x) is thrice differentiable function such that f(k)=-2, f(m)=-2 , f(l)=1,f(n)=3 and f(p)=-2,where k

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f

is continuous and differentiable for all

x > 0,

f^{2} (x) = x \int 0 f (t) \frac{cos t}{2 + sin t} d t

f (x) \neq 0, f (π) = ln 2,

f (x)

f : N \to N

be a function such that

f (m + n) = f (m) + f (n)

m, n \in N .

f (6) = 18,

f (2) \cdot f (3)

f : R \to R

is continuous and differentiable function such that

\int_{- 1}^{x} f (t) d t + f^{'''} (3) \int_{x}^{0} d t = \int_{1}^{x} t^{3} d t - f^{'} (t) \int_{0}^{x} t^{2} d t + f^{''} (2) \int_{x}^{3} t d t

f^{'} (4) = 48 - a f^{'} (1) - b f^{''} (2)

a + b =

f : R \to R

is a twice differentiable function such that

f " (x) > 0

\forall x ϵ R

f (\frac{1}{2}) = \frac{1}{2}, f (1) = 1

Q. Define a sequence

⟨ f_{0} (x), f_{1} (x), f_{2} (x), \dots ⟩

of functions by

f_{0} (x) = 1, f_{1} (x) = x, {(\begin{matrix} f_{n} (x) \end{matrix})}^{2} - 1 = f_{n + 1} (x) f_{n - 1} (x)

n \geq 1

Prove that each

f_{n} (x)

is a polynomial with integer coefficients.

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