Let $S$ be the set of all the natural numbers, for which the line $\frac{x}{a}+\frac{y}{b}=2$ is a tangent to the curve $\left(\frac{x}{a}\right)^n+\left(\frac{y}{b}\right)^n=2$ at the point...

$f(x)=4 \log _e(x-1)-2 x^2+4 x+5, x>1$, which one of the following is NOT correct?

Let the function $f(x)=\frac{x}{3}+\frac{3}{x}+3, x \neq 0$ be strictly increasing in $\left(-\infty, \alpha_1\right) \cup\left(\alpha_2, \infty\right)$ and strictly decreasing in $\left(\alpha_3, \alpha_4\right) \cup\left(\alpha_4, \alpha_5\right)$. Then $...

For the function $f(x)=(\cos x)-x+1, x \in \mathbb{R}$, between the following two statements (S1) $f(x)=0$ for only one value of $x$ is $[0, \pi]$. (S2)...

Let f(x)=x^5+2x^3+3x+1,x∈R, and g(x) be a function such that g(f(x))=x for all x∈R. Then (g(7))/(g^' (7)) is equal to :

For the function f(x)=sin⁡x+3x-2x2+x, where x0,2, consider the following two statements :
(I) f is increasing in 0,2.
(II) f' is decreasing in 0,2. Between...

Consider the function $\mathrm{f}:\left[\frac{1}{2}, 1\right] \rightarrow \mathrm{R}$ defined by $f(x)=4 \sqrt{2} x^3-3 \sqrt{2} x-1$. Consider the statements (I) The curve y=f(x) intersects the x-axis exactly...

Let $g: R \rightarrow R$ be a non constant twice differentiable such that $g^{\prime}\left(\frac{1}{2}\right)=g^{\prime}\left(\frac{3}{2}\right)$. If a real valued function $f$ is defined as $f(x)=\frac{1}{2}[g(x)+g(2-x)]$, then

Let $f(x)=x^3+x^2 f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime}(3), x \in R$. Then $f^{\prime}(10)$ is equal to