Optimization

Consider the vectors $a = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$, $b = \begin{bmatrix} 0 \\ 3\sqrt{2} \end{bmatrix}$. For real-valued scalar variable X, the value of $\min_{x} ||ax...

Consider the vectors $$ a=\left[\begin{array}{l} 1 \\ 1 \end{array}\right], b=\left[\begin{array}{c} 0 \\ 3 \sqrt{2} \end{array}\right] $$ For real-valued scalar variable $x$, the...

The random variable $X$ takes values in $\{-1,0,1\}$ with probabilities $P(X=-1)=P(X=1)$ and $\alpha$ and $P(X=0)=1-2 \alpha$, where $0 Let $g(\alpha)$ denote the entropy of $X$ (i...

Out of 1000 individuals in a town, 100 unidentified individuals are covid positive. Due to lack of adequate covid-testing kits, the health authorities of the town devised a strateg...

Out of 1000 individuals in a town, 100 unidentified individuals are covid positive. Due to lack of adequate covid-testing kits, the health authorities of the town devised a strateg...

The function f(x) = 8log e x $$-$$ x 2 + 3 attains its minimum over the interval [1, e] at x = __________. (Here log e x is the natural logarithm of x.)

Let $h[n]$ be a length-7 discrete-time finite impulse response filter, given by $h[0] = 4$, $h[1] = 3$, $h[2] = 2$, $h[3] = 1$, $h[-1] = -3$, $h[-2] = -2$, $h[-3] = -1$, and $h[n]$...

A connection is made consisting of resistance A in series with a parallel combination of resistances B and C. Three resistors of value 10 Ω, 5 Ω, 2 Ω are provided. Consider all pos...

The maximum area (in square units) of a rectangle whose vertices lie on the ellipse $${x^2} + 4{y^2} = 1\,\,$$ is

For a right angled triangle, if the sum of the lengths of the hypotenuse and a side is kept constant, in order to have maximum area of the triangle, the angle between the hypotenus...

The maximum value of $$f\left( x \right) = 2{x^3} - 9{x^2} + 12x - 3$$ in the interval $$\,0 \le x \le 3$$ is __________.

For $$0 \le t < \infty ,$$ the maximum value of the function $$f\left( t \right) = {e^{ - t}} - 2{e^{ - 2t}}\,$$ occurs at

The maximum value of the determinant among all $$2 \times 2$$ real symmetric matrices with trace $$14$$ is ______.

Consider the function $$\,f\left( x \right) = {x^2} - x - 2.\,$$ The maximum value of $$f(x)$$ in the closed interval $$\left[ { - 4,4} \right]\,$$

The function $$y = {x^2} + {{250} \over x}$$ at $$x=5$$ attains