Find the intervals where the following functions are increasing: (a) y = x² ln x (b) y = 4x − 5 ln(x² + 1) (c) y=3ln(1+x)+x−1/2x²

Question

Find the intervals where the following functions are increasing:

[latex](\mathbf{a})\ y=x^{2}\ln x\qquad\mathrm{(b)}\ y=4x-5\ln(x^{2}+1)\qquad(\mathbf{c})\ y=3\ln(1+x)+x-{\frac{1}{2}}x^{2}[/latex]

Accepted Answer

(a) The function is defined for x > 0, and

[latex]y^{\prime}=2x\ln x+x^{2}(1/x)=x(2\ln x+1)[/latex]

Hence, [latex] y^{\prime} ≥ 0[/latex] when ln x ≥ −1/2, that is, when [latex]x ≥ e^{−1/2}.[/latex] That is, y is increasing in the interval [latex][e^{−1/2},∞).[/latex]
(b) We find that

[latex]y^{\prime}=4-{\frac{10x}{x^{2}+1}}=4(x-2)\left(x-{\frac{1}{2}}\right)x^{2}+1[/latex]

A sign diagram reveals that y is increasing in each of the intervals [latex](−∞, \frac{1}{2}][/latex] and [latex][2,∞).[/latex]

(c) The function is defined for x > −1, and

[latex]y^{\prime}={\frac{3}{1+x}}+1-x={\frac{(2-x)(2+x)}{x+1}}[/latex]

A sign diagram reveals that y is increasing in (−1, 2].

Question 6.11.3: Find the intervals where the following functions are increas......

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