Expert answers to "One of the critical points of f(x,y)=x3−5y2+10y−27x is:..."

Phoebe

Answered question

2021-10-12

One of the critical points of

f (x, y) = x^{3} - 5 y^{2} + 10 y - 27 x

is:
(1,-1)
(-3,1)
(-3,-1)
None of these

Answer & Explanation

Obiajulu

Skilled2021-10-13Added 98 answers

Step 1
Necessary Condition for Critical Point:
A necessary condition for (x, y) to be a critical point of the function of two independent variables f(x, y) are
all partial derivatives $f_{x}, f_{y}$ exist and vanish at (x, y)
Step 2
The given function is
$f (x, y) = x^{3} - 5 y^{2} + 10 y - 27 x$ ...(1)
Now differentiating (1) partially with respect to x and y
we get
$f_{x} = 3 x^{2} - 27$ ....(2)
and $f_{y} = - 10 y + 10$ ...(3)
Step 3
For critical points, we must have $f_{x} = 0, f_{y} = 0$
Therefore,
$3 x^{2} - 27 = 0$
$3 x^{2} = 27$
$x^{2} = 9$
x=3, -3
and
-10y+10=0
-10y=-10
y=1
Hence critical points are (3, 1) & (−3, 1)
$∴$ One of the critical point of f(x, y) is (−3, 1)

Jeffrey Jordon

Expert2022-08-14Added 2605 answers

Answer is given below (on video)

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