Application Of Integration 1 over x
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The function \(f(x) = x\ln x\) has a derivative \(f'(x) = 1 + \ln x\) and a second derivative \(f''(x) = \dfrac{1}{x}\)
(i) Find the value of \(x\) for which \(y = f(x)\) has a stationary point
(ii) Find the value of \(x\) for which \(f(x)\) is increasing
(iii) Sketch the curve \(y = f(x)\) for \(0 \le x \le e\)
(iv) Describe the behaviour of the graph as \(x\) approaches 0
(i) \( - {e^{ - 1}}\)
(ii) Increasing for values of \(x > \frac{1}{e}\)
(iii) Refer to worked solutions
(iv) As \(x \to 0\) the function tends to the origin, however is never equal to 0 since \(\ln 0\) is not defined.
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