---
# System prepended metadata

title: Machine Learning Exercise 1

---

# Machine Learning Exercise 1

## Linear Algebra

### Problem 1

$$ A \in R^{M \times N}, \quad b \in R^{1 \times N},  $$

## Calculus 

### Problem 7

$$f(x) = \frac{1}{2}ax^2+bx+c$$
$$f'(x) = ax + b$$
$$f''(x) = a$$

a)
(i) $f(x)$ has only one solution if $a>0$ because $f'(x) = 0$ has only a solution and $f''(x)$ is positive (it is a point of minimum).
(ii) $f(x)$ has infinite solutions if $a=0 \land b=0$ because $f'(x)=0 \quad \forall x$
(iii) 
- if $a=0 \land b\neq 0$ there is no solution because $\nexists x | f'(x) = 0$ 
- if $a<0$ there is no solution because $f''(x) < 0 \quad \forall x$ and that means that $f'(x)$ is a point of maximum.  

b) Assuming that the optimization problem has a unique solution, the point of minimum is where $$f'(x)= ax+b=0$$ and it holds for $$x* = argmin_{x \in R} f(x)= -\frac{b}{a}$$ (knowing that $a>0$ by previous exercise)
