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In which interval is the given function $f(x)=2 x^3-15 x^2+36 x+1$ is monotonically decreasing
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Verified Answer
The correct answer is:
$(2,3)$
$\begin{aligned}& y=f(x)=2 x^3-15 x^2+36 x+1 \\
& \frac{d y}{d x}=f^{\prime}(x)=6 x^2-30 x+36=6\left(x^2-5 x+6\right) \\
& f^{\prime}(x)=6(x-2)(x-3)\end{aligned}$
To be monotonic decreasing, $f^{\prime}(x) \lt 0$
$\Rightarrow(x-2)(x-3) \lt 0 \Rightarrow x \in(2,3)$
& \frac{d y}{d x}=f^{\prime}(x)=6 x^2-30 x+36=6\left(x^2-5 x+6\right) \\
& f^{\prime}(x)=6(x-2)(x-3)\end{aligned}$
To be monotonic decreasing, $f^{\prime}(x) \lt 0$
$\Rightarrow(x-2)(x-3) \lt 0 \Rightarrow x \in(2,3)$
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