Application of Derivatives

Let $f(x)=\ln x-2023 x+2023$ for all $x \in(0, \infty)$. Then the number of points at which the graph of $f$ cuts the $x$ axis is

For $a, b \in \mathbb{R}$, with $a \gt 0$, let $N(a, b)$ denote the number of elements in the set $\{x \in \mathbb{R} \mid x+a \sin x=b\}$. Then

Let $f$ be the function on $\mathbb{R}$ defined by $f(x)=x^3-3 x^2+a x-1$, where $a \in \mathbb{R}$. Then the set of all possible values of $a$ for which $f$ is strictly increasing is

Let $f:(0,3) \cup(6,9) \rightarrow \mathbb{R}$ be a differentiable function such that $f^{\prime}(x)=\frac{1}{2}$ for all $x \in(0,3) \cup(6,9)$. Then

Let $g: \mathbb{R} \mapsto \mathbb{R}$ be a differentiable function such that $g(x) g^{\prime}(x) \gt 0$ for all $x \in \mathbb{R}$. Then

Let $a$ and $b$ be non-negative real numbers satisfying $a^2+b^2=4$. Then the minimum value of $4^a+4^b$ is

The number of solutions of $e^{-x}=\sin x$ is

A continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ is said to be good if given $a, b \in \mathbb{R}(a \lt b)$, the line segment joining the points $(a, f(a))$ and $(b, f(b))$ lies on or above the graph of $f$…

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $\lim _{x \rightarrow \infty} \frac{f(x)}{x}=0$. Then

Let $P$ be a polynomial whose coefficients are real numbers. Suppose the roots of $P(x)=0$ are real. Then

Consider the equations $y=x^3-x^2+3 x-4$ and $y=\alpha x^2-x-4, \quad \alpha \in \mathbb{R.}$ The number of values of $\alpha$ for which the above two equations intersect at exactly two points is

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a nonconstant differentiable function such that $\lim _{x \rightarrow \infty} f(x)=1$ and $\lim _{x \rightarrow \infty} f^{\prime}(x)=\alpha$ for some $\alpha \in…