Find 𝑓𝑥𝑥 , 𝑓𝑥𝑦 and 𝑓𝑦𝑦 for the function 𝑓(𝑥,𝑦)=3𝑥2+3𝑥2𝑦2+2𝑥𝑦4+3𝑦3 Question 6Select one: a. 𝑓𝑥𝑥=6+12𝑦2,𝑓𝑥𝑦=12𝑥𝑦+12𝑦3,𝑓𝑦𝑦=3𝑥2+24𝑥𝑦2+12𝑦. b. 𝑓𝑥𝑥=12+6𝑦2,𝑓𝑥𝑦=6𝑥𝑦+12𝑦3,𝑓𝑦𝑦=12𝑥2+12𝑥𝑦2+18𝑦. c. 𝑓𝑥𝑥=6+6𝑦2,𝑓𝑥𝑦=12𝑥𝑦+8𝑦3,𝑓𝑦𝑦=6𝑥2+24𝑥𝑦2+18𝑦. d. 𝑓𝑥𝑥=6+6𝑦2,𝑓𝑥𝑦=8𝑥𝑦+12𝑦3,𝑓𝑦𝑦=6𝑥2+12𝑥𝑦2+12𝑦. e. None of these

To find the second partial derivatives of the function 𝑓(𝑥,𝑦)=3𝑥²+3𝑥²𝑦²+2𝑥𝑦⁴+3𝑦³, we first need to find the first partial derivatives.

The partial derivative of f with respect to x, denoted 𝑓𝑥, is found by differentiating f with respect to x while treating y as a constant:

𝑓𝑥 = 6𝑥 + 6𝑥𝑦² + 2𝑦⁴.

The partial derivative of f with respect to y, denoted 𝑓𝑦, is found by differentiating f with respect to y while treating x as a constant:

𝑓𝑦 = 6𝑥²𝑦 + 8𝑥𝑦³ + 9𝑦².

Now we can find the second partial derivatives. The second partial derivative of f with respect to x, denoted 𝑓𝑥𝑥, is found by differentiating 𝑓𝑥 with respect to x:

𝑓𝑥𝑥 = 6 + 6𝑦².

The second partial derivative of f with respect to y, denoted 𝑓𝑦𝑦, is found by differentiating 𝑓𝑦 with respect to y:

𝑓𝑦𝑦 = 6𝑥² + 24𝑥𝑦² + 18𝑦.

The mixed second partial derivative of f, denoted 𝑓𝑥𝑦, is found by differentiating 𝑓𝑥 with respect to y:

𝑓𝑥𝑦 = 12𝑥𝑦 + 8𝑦³.

So, the correct answer is not listed among the options. The correct second partial derivatives are 𝑓𝑥𝑥=6+6𝑦², 𝑓𝑥𝑦=12𝑥𝑦+8𝑦³, 𝑓𝑦𝑦=6𝑥²+24𝑥𝑦²+18𝑦. Therefore, the answer is e. None of these.