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Solution to Mathematics for Machine Learning Exercise 7.1


Consider the univariate function $$f(x)=x^3+6x^2-3x-5.$$Find its stationary points and indicate whether they are maximum, minimum, or saddle points.


I will assume you are familiar with Calculus, see the book Calculus for more details.

Solution: First, we compute the derivative of $f(x)$, $$f'(x)=3x^2+12x-3=3(x^2+4x-1).$$Its stationary points (also known as critical points) are the zeros of derivative. Hence by quadratic formula, we see that the stationary points are $$x_1=-2+\sqrt{5},\quad x_2=-2-\sqrt{5}.$$To determine the local extreme values, we have find the signs of the derivative on some intervals. Since we know that the graph of the derivative is an open-upward parabola with $x$-intercepts $-2\pm \sqrt 5$, it is clear that the function $f'(x)$ is positive on $(-\infty,-2-\sqrt 5)$ and $(-2+\sqrt 5,\infty)$ and negative on $(-2-\sqrt 5,-2+\sqrt 5)$. Below is the graph of the derivative.

Solution to Mathematics for Machine Learning Exercise 7.1

Hence by the first derivative test, we see that the original function has local maximum at $x_2=-2-\sqrt{5}$ and local minimum at $x=-2+\sqrt{5}$. There is no global maximum or global minimum since $f(-\infty)=-\infty$ and $f(\infty)=\infty$. Here is the graph of the original function.

Solution to Mathematics for Machine Learning Exercise 7.1

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