The function f(x) = 2x + 3x^{frac{2}{3}}, x i | vedclass

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(C) Given function is $f(x) = 2x + 3x^{\frac{2}{3}}$.First,find the derivative $f'(x)$:$f'(x) = 2 + 3 \cdot \frac{2}{3} x^{\frac{2}{3} - 1} = 2 + 2x^{

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exactly one point of local maxima and exactly one point of local minima

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(C) Given function is $f(x) = 2x + 3x^{\frac{2}{3}}$.First,find the derivative $f'(x)$:$f'(x) = 2 + 3 \cdot \frac{2}{3} x^{\frac{2}{3} - 1} = 2 + 2x^{-\frac{1}{3}} = 2 + \frac{2}{x^{\frac{1}{3}}} = 2 \left( \frac{x^{\frac{1}{3}} + 1}{x^{\frac{1}{3}}} \right)$.Critical points occur where $f'(x) = 0$ or $f'(x)$ is undefined.$f'(x) = 0 \implies x^{\frac{1}{3}} + 1 = 0 \implies x = -1$.$f'(x)$ is undefined at $x = 0$.Now,check the sign of $f'(x)$ around these points:For $x  0$.For $-1 For $x > 0$,$f'(x) > 0$.Since $f'(x)$ changes from positive to negative at $x = -1$,there is a local maximum at $x = -1$.Since $f'(x)$ changes from negative to positive at $x = 0$,there is a local minimum at $x = 0$.Thus,the function has exactly one point of local maxima and exactly one point of local minima.

The function $f(x) = 2x + 3x^{\frac{2}{3}}, x \in R$,has

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Statement-$I$: The sequence $a_n = \frac{n^2}{n^3 + 200}, n \in N$ has its $7^{th}$ term as the largest term.
Statement-$II$: The function $f(x) = \frac{x^2}{x^3 + 200}$ attains a local maximum at $x = 7$.

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