Question

Comsider this, multivariable fucntion f(x,y)=3xy-x^2+5y^2-25 a)What is the value of f(-1,3)? b)Find all x-values such that f(x,x)=0

Multivariable functions
ANSWERED
asked 2020-11-27
Comsider this, multivariable fucntion
\(\displaystyle{f{{\left({x},{y}\right)}}}={3}{x}{y}-{x}^{{2}}+{5}{y}^{{2}}-{25}\)
a)What is the value of f(-1,3)?
b)Find all x-values such that f(x,x)=0

Expert Answers (1)

2020-11-28
a) Here \(\displaystyle{f{{\left({x},{y}\right)}}}={3}{x}{y}-{x}^{{2}}+{5}{y}^{{2}}-{25}\) (1)
put x=-1, y=3 in (1), we get
\(\displaystyle{f{{\left(-{1},{3}\right)}}}={3}\cdot{\left(-{1}\right)}\cdot{3}-{\left(-{1}\right)}^{{2}}+{5}\cdot{\left({3}\right)}^{{2}}-{25}\)
=-9-1+15-25
=-10+15-25
=-10-10
=-20
Hence f(-1,3)=-20
b) Now \(\displaystyle{f{{\left({x},{y}\right)}}}={3}{x}{y}-{x}^{{2}}+{5}{y}^{{2}}-{25}={0}\)
\(\displaystyle\Rightarrow{3}{x}^{{2}}-{x}^{{2}}+{5}{x}^{{2}}-{25}={0}\)
\(\displaystyle\Rightarrow{7}{x}^{{2}}={25}\)
\(\displaystyle\Rightarrow{x}^{{2}}=\frac{{25}}{{7}}\)
\(\displaystyle\Rightarrow{x}=\pm\sqrt{{\frac{{25}}{{7}}}}\)
\(\displaystyle\Rightarrow{x}=\pm\frac{{5}}{\sqrt{{7}}}\)
Hence \(\displaystyle{x}=\frac{{5}}{\sqrt{{7}}},-\frac{{5}}{\sqrt{{7}}}\)
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