# Worksheet 2-1

Download: [CMSE382-WS2_1.pdf](CMSE382-WS2_1.pdf)

```{warning}
This is an AI-generated transcript of the worksheet and may contain errors or inaccuracies. Please refer to the original course materials for authoritative content.
```

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## Worksheet 2-1: Q1

The function $f(x)=3x^4 - 28x^3 + 84x^2 - 96x$ over $[-1,6]$ is plotted at [desmos.com/calculator/nevlfz0yxp](https://www.desmos.com/calculator/nevlfz0yxp). The derivative of $f$ is $f'(x) = 12x^3 - 84x^2 +168x-96= 12(x-1)(x-2)(x-4)$.

- Using the derivative, find all critical points of $f(x)$.

- Identify all stationary points, and for each stationary point, identify its type (min, max, saddle).

- Find the global minimum and maximum of $f(x)$ over the interval $[-1,6]$.

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## Worksheet 2-1: Q2

Find the stationary points of $f(x,y)=6x^2 y - 3x^3 + 2y^3 - 150y$.

*(You can visually check your answer at [desmos.com/3d/xa4komuwmb](https://www.desmos.com/3d/xa4komuwmb).)*

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## Worksheet 2-1: Q3

On a quiz, Dr. Munch asks about a function $f:U \to \mathbb{R}$ defined on $U \subseteq \mathbb{R}^n$ where all partial derivatives of $f$ exist.

- **Kylo Ren** writes: *"At a local optimum, the gradient is zero, so $\nabla f(x^*)=0$."*

  Mark his answer correct or explain why his answer is wrong.

- **Rey Skywalker** writes: *"Since $\nabla f(\mathbf{x}^*)=0$, $f$ has a local optimum."*

  Mark her answer correct or explain why her answer is wrong.

- **Poe Dameron** writes: *"Since $f$ has a local optimum at $\mathbf{x}^*$ and $\mathbf{x}^*$ is in the interior of $U$, the gradient is zero, meaning $\nabla f(\mathbf{x}^*)=0$."*

  Mark his answer correct or explain why his answer is wrong.

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## Worksheet 2-1: Q4

Let $f(x,y)=2x+3y:S\to \mathbb{R}$ and $S=B[0,1]=\{(x,y): x^2+y^2 \leq 1\}$.

(a) We call $f$ a linear map if there is a matrix $A$ such that $f(x,y)=A \mathbf{x}$, where $\mathbf{x}=\begin{bmatrix} x\\ y \end{bmatrix}$. Find the matrix $A$ to show that $f$ is a linear map.

(b) Note that because $A$ is just a vector, $A\mathbf{x}$ is the same as the dot product. Use the Cauchy-Schwarz inequality to find $\argmin_{x \in S} f(x)$ and $\argmax_{x \in S} f(x)$.

(c) What do the points you found in part (b) represent?