For any twice differentiable function f : R → R, if at some x* ∈ R, f'(x*) = 0 and f''(x*) > 0, then the function f necessarily ha…

Let \(f:\mathbb{R}\to\mathbb{R}\) be the function \(f(x)=\frac{1}{1+e^{-x}}\). The value of \(f'(x)\) at a point where \(f(x)=0.4\…

Let \(f:\mathbb{R}\to\mathbb{R}\) be defined by \[ f(x)=\begin{cases} -x, & x<-2,\\ ax^2+bx+c, & -2\le x\le2,\\ x, & x>2. \end{ca…

Evaluate the limit \[ \lim_{x\to0}\frac{\ln\!\left((x^2+1)\cos x\right)}{x^2}=\boxed{\phantom{0}}. \]

Let \(f(x) = x^3 - 3x^2 + 2\) be a function defined on \((-1, 3]\). Which of the following statements is/are correct?

Consider that for a supervised learning task, the objective function being minimized is \(f_w(x) = wx\), where \(x \in \mathbb{R}\…