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Which of the following statement(s) is/are TRUE?A. If function `y=f(x)` is continuous at `x=c` such that `f(c)!=0`, then `f(x)f(c)gt0AA x epsilon(c-h,c+h)`, where `h` is sufficiently small positive quantity.B. `lim_(n to oo) 1/n (n (1+1/n)(1+2/n)……(1+n/n))=1+2In2`.C. Let `f` be a continuous and non-negative function defined on`[a,b]` If `int_(a)^(b)f(x)dx=0`, then `f(x)=AA x epsilon [a,b]`D. Let `f` be continuous function defined on `[a,b]` such that `int_(a)^(b)f(x)dx=0`.Then there exists at least one `c epsilon(a,b)` for which `f(c)=0` |
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Answer» Correct Answer - A::C::D (1) The expression `f(x)f(c)AAx epsilon(c-h,c+h)` where `hto0^(+)` is equivalent to `lim_(xtoc)f(x)f(c)`which is equal to `f(c))^(2)` because `f(x)` is continuous. There `f(x)f(c)gt0AA(c-h,c+h)` where `hto0^(+)`. (2) We have `I=lim_(nto oo) 1n In[(1+1/n)(1+2/n)........(1+n/n)]` `=lim_(nto oo) 1/n In prod_(k=1)^(n)(1+k/n)` `=lim_(nto oo) 1/n sum_(k=1)^(n)In(1+k/n)` `=int_(0)^(1)log_(e)(1+x)dx` `=int_(1)^(2)Inx dx=[(Inx-1)]_(1)^(2)=-1+2In2` ltbgt (3) Given `f(x)gt0` or `int_(a)^(b)f(x)dxge0`. But given `int_(a)^(b)f(x)dx=0`. So, this can be true only when `f(x)=0` (4) `int_(a)^(b)f(x)dx+0` i.e. `y=f(x)` cuts x-axis at least once. |
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