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Graphical Interpretation of Derivative

f(x) is a di...

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f(x) is a differentiable function and g(x) is a double differentiable function such that |f(x)|≤1 and f'(x)=g(x) . If f2(0)+g2(0)=9 . Prove that there exists some c∈(–3,3) such that g(c).g''(c)<0.

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Q. If f(x) and g(x) are differentiable functions in [0,1] such that f(0)=2=g(1), g(0)=0, f(1)=6 then there exists c,

0 < c < 1

such that f '(c) =

Q. Suppose that

f (x)

is a differential function such that

f^{'} (x)

f^{'} (0) = 1

f^{'} (0)

does not exist. Let

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

g (x) = \frac{f (x)}{x + 1}

f (x)

is differentiable on

[0, 5]

f (0) = 4, f (5) = - 1

c \in (0, 5)

g^{^{'}} (c)

Q. Let f be a twice differentiable function such that

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

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

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

h (1) = 8 a n d h (0) = 2,

h (2)

f

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 (0) = 2

h (2) =

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