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Differentiability

APPLICATION O...

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APPLICATION OF DERIVATIVES : Integer Type Let f(x) be a non-constant thrice differentiable function defined on R such that f(x) = f(6-x) and p(0)=0=p(2)=p(5).If n is the minimum number of roots of (g(x)) + p(x) h(x) =0 in the interval [0,6] then the value of n/2 is? p, g, h are the first, second and third derivatives of f(x) w.r.t. x.

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f (x)

be a non-constant, thrice differentiable function defined on

R

f (x) = f (6 - x)

f^{'} (0) = 0 = f^{'} (2) = f^{'} (5) .

n

is the minimum number of roots of

{(f^{'} (x))}^{2} + f^{'} (x) \cdot f^{'''} (x) = 0

in the interval

[0, 6]

, then the value of

\frac{n}{2}

Q. Assertion(A): Let

f (x)

be twice differentiable function such that

f^{^{''}} (x) = - f (x)

f^{^{'}} (x) = g (x)

h (x) = [f (x)]^{2} + [g (x)]^{2}

h (1) = 8

h (2) = 8

Reason (R): Derivative of a constant function is zero.

y = f (x)

be a thrice differentiable function defined on R such that f(x) = 0 has atleast 7 distinct zeros, then minimum number of zeros of the equation

f (x) + 9 f^{^{'}} (x) + 27 f^{^{''}} (x) + 27 f^{^{'''}} (x) = 0

f (x)

be a non-positive continuous function and

F (x) = x \int 0 f (t) d t \forall x \geq 0

f (x) \geq c F (x)

c > 0

g : [0, \infty) \to R

be a function such that

\frac{d g (x)}{d x} < g (x) \forall x > 0

g (0) = 0.

The total number of root(s) of the equation

f (x) = g (x)

f

be a non-negative continuous and twice differentiable function defined on

R

f (x) + f (x + \frac{3}{2}) = 6, \forall x \in R .

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