Find the intervals in which function f(x)=4 x^3-6 x^2-72 x+3 is m… - Vidyadip

Question

Find the intervals in which function $f(x)=4 x^3-6 x^2-72 x+3$ is mcreasing.

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Answer

Find the derivative $f^{\prime}(x): f^{\prime}(x)=12 x^2-12 x-72=12\left(x^2-x-6\right)$
Factor the quadratic: $f^{\prime}(x)=12(x-3)(x+2)$
For the function to be increasing, $f^{\prime}(x) \geq 0$.
Check intervals on a number line with critical points $x=-2$ and $x=3$ :
- $(-\infty,-2]$ : $f^{\prime}(x)$ is positive (+).
- $[-2,3]: f^{\prime}(x)$ is negative (-).
- $[3, \infty): f^{\prime}(x)$ is positive (+). Final Answer: The function is increasing in the intervals $(-\infty,-2] \cup[3, \infty)$.

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