AMU MCA Maxima And Minima Previous Year Questions (PYQs)
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Showing 5 questions
The function $f(x,y) = 2x^4 - 3x^2y + y^2$ has:
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Solution
At $(0,0)$, first partial derivatives vanish and second derivative test confirms a minimum.
The sum of two numbers is $10$. Their product will be maximum when they are
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Solution
Let numbers be $x$ and $10-x$
Product $= x(10-x) = 10x - x^2$
Maximum at vertex: $x = \frac{10}{2} = 5$
Numbers: $5,5$
The maximum value of $\frac{\log x}{x}$ is
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Solution
Let $f(x)=\frac{\log x}{x}$
$f'(x)=\frac{1-\log x}{x^2}$
Set $f'(x)=0 \Rightarrow \log x=1 \Rightarrow x=e$
Maximum value $= \frac{\log e}{e}=\frac{1}{e}$
The greatest value of $f(x,y)=xy$ on ellipse
$ \frac{x^2}{8}+\frac{y^2}{2}=1 $
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Solution
Using Lagrange multipliers, maximum occurs at
$ x=2,\ y=1 $
So
$ xy=2 $
The minimum value of $9^x + 9^{1-x}$, $x \in R$ is
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Solution
Let $a=9^x$ ⇒ expression $= a + \frac{9}{a}$
Minimum at $a=3$
Value $= 3 + 3 = 6$
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