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Question

Suppose that f is differentiable function with the property f(x+y)=f(x)+f(y)+x2y2 and limx0f(x)x=10 then f(0) is equal to

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Solution

The derivative of a function using the first principle is when L.H.D is equal to the R.H.D.

R.H.D. f(x)=limh0f(x+h)f(x)h

L.H.D. f(x)=limh0f(xh)f(x)h

Evaluating the above we get,

R.H.D.

f(0)=limh0f(0+h)f(0)h

f(0)=limh0f(0)+f(h)+0.h2f(0)h

f(0)=limh0f(h)h

f(0)=10

L.H.D.

f(0)=limh0f(0h)f(x)h

f(0)=limh0f(0)+f(h)+0.h2f(0)h

f(0)=limh0f(h)h

f(0)=10

Thus L.H.D. = R.H.D. = 10 .....Answer


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