Find all the points of local maxima and local | vedclass

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(D) Given function is $f(x) = 2x^3 - 6x^2 + 6x + 5$. First,we find the derivative $f'(x)$: $f'(x) = \frac{d}{dx}(2x^3 - 6x^2 + 6x + 5) = 6x^2 - 12x +

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No local maxima and no local minima

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(D) Given function is $f(x) = 2x^3 - 6x^2 + 6x + 5$. First,we find the derivative $f'(x)$: $f'(x) = \frac{d}{dx}(2x^3 - 6x^2 + 6x + 5) = 6x^2 - 12x + 6$. Setting $f'(x) = 0$ to find critical points: $6(x^2 - 2x + 1) = 0$ $6(x - 1)^2 = 0$ This gives $x = 1$ as the only critical point. Now,we examine the sign of $f'(x)$ around $x = 1$: For $x  0$. For $x > 1$,let $x = 2$,then $f'(2) = 6(2-1)^2 = 6 > 0$. Since $f'(x)$ does not change sign as $x$ passes through $1$ (it remains positive),$x = 1$ is neither a point of local maxima nor a point of local minima. Therefore,the function has no local maxima and no local minima.

Find all the points of local maxima and local minima of the function $f$ given by $f(x) = 2x^3 - 6x^2 + 6x + 5$.

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