Question

Math

Posted 6 months ago

```
$f(x, y)=\frac{x^{2}}{2}-x^{4}+\cos (x)-y^{2}+2 y$
What are all the critical points of $f$ ?
Choose 1 answer:
(A) $(-0.511 \ldots, 1)$
(B) $(0,1)$
(C) $(0.511 \ldots, 1)$
(D) There are no critical points.
```

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Answer from Sia

Posted 6 months ago

Solution by Steps

step 2

The partial derivative with respect to $x$ is $f_x(x, y) = x - 4x^3 + \sin(x)$. Setting this equal to zero gives us the equation $x - 4x^3 + \sin(x) = 0$

step 3

The partial derivative with respect to $y$ is $f_y(x, y) = -2y + 2$. Setting this equal to zero gives us $-2y + 2 = 0$, which simplifies to $y = 1$

step 4

Substituting $y = 1$ into $f_x(x, y) = 0$, we need to solve $x - 4x^3 + \sin(x) = 0$ for $x$. This is a transcendental equation and may have multiple solutions

step 5

The asksia-ll calculator has provided the solutions to the equation $x - 4x^3 + \sin(x) = 0$ as $x \approx -0.511...$ and $x \approx 0.511...$

step 6

Therefore, the critical points of the function are approximately $(-0.511..., 1)$ and $(0.511..., 1)$

[question 1] Answer

A and C

Key Concept

Critical Points of a Multivariable Function

Explanation

Critical points occur where the gradient of the function is zero. For a function $f(x, y)$, this means solving $f_x(x, y) = 0$ and $f_y(x, y) = 0$ simultaneously.

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