Determine the local maximum and local minimum points for the function $f(x) = 2x^3 + 3x^2 – 12x + 3$.
I know that the local maximum and local minimum points largest and smallest value of the function, but I do not understand how to find the points from the equation. Assistance would be greatly appreciated.
Best Answer
To find the critical points of $f$, you should differentiate and set the derivative equal to 0.
$f(x) = 2x^3 + 3x^2 - 12x + 3 \implies f'(x) = 6x^2 + 6x - 12$
Then, $f'(x) = 0 \iff 6x^2 + 6x- 12 = 0 \iff x^2 +x -2$.
Using Bhaskara, you find the roots of this polynomial ($-2$ and $1$).
Now evaluate $f(-2)$ and $f(1)$ to decide which one is the maximum and which one is the minimum.
You should also note that
$\lim_{x \rightarrow -\infty} f(x) = -\infty$ and $\lim_{x \rightarrow +\infty} f(x) = +\infty$