The function f(x)=2 x^3-9 x^2+12 x+29 is mono | vedclass

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(A) Given function: $f(x)=2 x^3-9 x^2+12 x+29$To find the intervals where the function is monotonically increasing,we find the derivative $f^{\prime}(

Vedclass · Accepted Answer

$(-\infty, 1) \cup(2, \infty)$

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(A) Given function: $f(x)=2 x^3-9 x^2+12 x+29$To find the intervals where the function is monotonically increasing,we find the derivative $f^{\prime}(x)$:$f^{\prime}(x) = \frac{d}{dx}(2 x^3-9 x^2+12 x+29) = 6 x^2-18 x+12$Factor the derivative:$f^{\prime}(x) = 6(x^2-3 x+2) = 6(x-1)(x-2)$For the function to be monotonically increasing,we require $f^{\prime}(x) > 0$:$6(x-1)(x-2) > 0$Using the sign scheme method on the number line with critical points $x=1$ and $x=2$:- For $x  0$ (positive)- For $1 - For $x > 2$,$f^{\prime}(x) > 0$ (positive)Thus,$f(x)$ is monotonically increasing in the interval $(-\infty, 1) \cup(2, \infty)$.

The function $f(x)=2 x^3-9 x^2+12 x+29$ is monotonically increasing in the interval

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