## Related questions with answers

Sketch the graphs of the given functions by determining the appropriate information and points from the first and second derivatives. Use a calculator to check the graph.

$y = x^3 - 12x + 12$

Solution

VerifiedIn this problem, our goal is to sketch the graph for the polynomial function $x^3-9x^2+15x+1$

To decide the increasing/decreasing nature and the concavity, we will need the first two derivatives of $f(x)$

$\begin{aligned} f'(x)&=\dfrac{d\left(x^3-12x+12\right)}{dx}\\ &=\dfrac{d\left(x^3\right)}{dx}-\dfrac{d\left(12x\right)}{dx}+\dfrac{d\left(12\right)}{dx},&&\text{Use the Sum/Difference Rule}\\ &=3x^{3-1}-12\cdot1+0,&&\text{Use the Power Rule}\\ &=3x^2-12\\ f''(x)&=\dfrac{d\left(3x^2-12\right)}{dx},&&\text{Again, Differentiate}\\ &=\dfrac{d\left(3x^2\right)}{dx}-\dfrac{d\left(12\right)}{dx},&&\text{Use the Difference Rule}\\ &=3\cdot2x^{2-1}-0,&&\text{Use the Power Rule}\\ &=6x \end{aligned}$

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