Recent questions tagged differentiation

​​​​​Let $f(x)=\frac{e^{x}-e^{-x}}{2}, x \in \mathbb{R}$. Let $f^{(k)}(a)$ denote the $k^{t h}$ derivative of $f$ evaluated at $a$. What is the value of $f^{(10)}(0)$? (N...

Consider two functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow(1, \infty)$. Both functions are differentiable at a point $c$. Which of the f...

​​​​Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice-differentiable function and suppose its second derivative satisfies $f^{\prime \prime}(x)>0$ for all $x \in \mat...

Consider the given function $f(x)$.$$f(x)=\left\{\begin{array}{ll} a x+b & \text { for } x<1 \\x^{3}+x^{2}+1 & \text { for } x \geq 1\end{array}\right.$$If the function i...

​​​​​Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. Note: $\mathbb{R}$ denotes the set of real numbers.\[f(x)=\left\{\begin{array}{cl}-x, & \text { if } x&lt-2...

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x)=\max \left\{x, x^3\right\}, x \in \mathbb{R}$, where $\mathbb{R}$ is the set of all real numbers....