Tag: second derivative test

Questions Related to second derivative test

Let $f(x) = ax^2+bx+c, a, b, c \in R.$ It is given $|f(x)| \le 1, \, |x| \le 1$ then the possible value of $\dfrac{8}{3}a^2+2b^2$ is given by

maths application of derivatives - iii second derivative test maxima and minima application of derivatives

  1. $32$

  2. $\dfrac{32}{3}$

  3. $\dfrac{2}{3}$

  4. $\dfrac{16}{3}$

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Correct Option: A

Let $x$ and $y$ be two positive real numbers such that $xy = 1.$ The minimum value of $x + y$ is

maths application of derivatives - iii second derivative test maxima and minima application of derivatives

  1. $1$

  2. $1/2$

  3. $2$

  4. $1/4$

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Correct Option: C
Explanation:

Given $xy=1$ and $f(x,y)=x+y$
$\Rightarrow f(x)=x+\dfrac{1}{x}$
$f'(x)=1-\dfrac{1}{x^2}$
For maxima or minima,
$f'(x)=0$
$\Rightarrow x=\pm1$
$f''(x)=\dfrac{2}{x^3}$
$f''(x)>0$ at $x=1$
Hence f(x) has minimum at $x=1$
$f(1)=2$
So, minimum value of $x+y  \ is  \  2$.