Define a function \(f:[0, 1]\to\mathbf{R}\) as \(f(x)=x^{2}+e^{x}-2\). Then :
\(f\) is increasing
Differentiate: \(f'(x)=2x+e^{x}\). On \([0,1]\) both terms are non-negative and \(e^{x}\geq 1\), so \(f'(x)\geq 1\gt 0\) throughout the interval.
A function with positive derivative on an interval is strictly increasing there (mean value theorem: \(f(y)-f(x)=f'(\xi)(y-x)\gt 0\) for \(y\gt x\)).
Since \(f'\) never vanishes, \(f'(x)=0\) has no solution in \([0,1]\); and a strictly increasing function is one-one, so the claim that it is not one-one is false.
The equation \(f(x)=0\) does have a solution: \(f(0)=-1\lt 0\) and \(f(1)=e-1\gt 0\), so by the intermediate value theorem there is a root in \((0,1)\).
Check: \(f(0.5)\approx 0.25+1.6487-2=-0.101\) and \(f(0.6)\approx 0.36+1.8221-2=0.182\), so the values rise through \(0\) near \(x\approx 0.54\), consistent with a strictly increasing function.
Hence, \(f\) is increasing.
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\(\frac{2}{1} + \frac{3}{4} + \frac{4}{9} + \ldots + \frac{n+1}{n^2} + \ldots\)
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\(f_n(x) = \frac{x^n}{1 + x^n}, \ 0 \le x \le 1.\)
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\(S_1 : f_n \to f \equiv 0\) uniformly on \([0, 1]\)
\(S_2 : \sup_{0 \le x \le \frac{1}{2}} f_n(x) \to F(x) \equiv 0\) on \([0, \frac{1}{2}]\)
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\(f(x, y) = \begin{cases} \frac{xy}{x^2 + y^2}, & (x, y) \neq (0, 0) \\ 0, & (x, y) = (0, 0) \end{cases}\)
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Important Questions from Mixed Topic (MH-SET)
.................... is a client server application that identifies each host on the internet with a unique user-friendly name.
Every OS has a ................ or .......... which primarily resides in the main memory of the computer to perform some of the basic functions of the OS.
(i) Kernel
(ii) Interface
(iii) Nucleus
(iv) Compiler
Codes:
Which of the following are referred to as low level language?
(i) Machine language
(ii) Assembly language
(iii) High level language
(iv) Natural language
Codes:
Match the following:
List I List II (a) New York Public Library (i) Shelv AR (b) Miami University Augmented Research Group (ii) Smart library (c) Oula University Library (iii) Wolf Walk (d) North Carolina State University (iv) Find the future Codes:
Which part of the QR code represents the encoded data?
