Tag: maxima and minima

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Questions Related to maxima and minima

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

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

  1. $32$
  2. $\dfrac{32}{3}$
  3. $\dfrac{2}{3}$
  4. $\dfrac{16}{3}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the constraints |ax^2+bx+c| <= 1 for |x| <= 1, the maximum value of the quadratic form 8/3*a^2 + 2*b^2 is derived from the bounds on coefficients of polynomials bounded by 1 on [-1, 1]. The calculated maximum is 32.

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

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

  1. $1$
  2. $1/2$
  3. $2$
  4. $1/4$
Reveal answer Fill a bubble to check yourself
C Correct answer
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 $x+y  \ is  \  2$.