Application of Derivatives

Let $f(x)=\frac{x}{\ln x} ;(x \gt 1)$ Consider the following statements: $f(x)$ is increasing in the interval $(e, \infty)$ $f(x)$ is decreasing in the interval $(1, e)$ $9 \ln 7 \gt 7 \ln 9$ Which of the statements…

If $f(x)=\frac{x^3}{3}-\frac{5 x^2}{2}+6 x+7$ increases in the interval T and decreases in the interval S , then which one of the following is correct?

Paragraph: Three sides of a trapezium are each equal to 6 cm . Let $\alpha \in\left(0, \frac{\pi}{2}\right)$ be the angle between a pair of adjacent sides. Question: If the area of the trapezium is the maximum possible,…

In which one of the following intervals is the function $f(x)=\frac{x^3}{3}-\frac{7 x^2}{2}+6 x+5$ decreasing?

Paragraph: Consider the function $f(x)=3 x^4-20 x^3-12 x^2+288 x+1$ Question: In which one of the following intervals is the function decreasing?

Passage: Let $S$ and $T$ be the sets where $f(x)=\dfrac{x^3}{3}-\dfrac{5x^2}{2}+6x+7$ decreases and increases respectively. Question: What is $T$ equal to ?

Let $f(x)=\frac{x}{\ln x} ;(x \gt 1)$ Consider the following statements: $\quad f^{\prime \prime}(e)=\frac{1}{e}$ $\mathrm{f}(\mathrm{x})$ attains local minimum value at $x=e$ A local minimum value of $f(x)$ is $e$…

Passage: Let $S$ and $T$ be the sets where $f(x)=\dfrac{x^3}{3}-\dfrac{5x^2}{2}+6x+7$ decreases and increases respectively. Question: What is $S$ equal to ?

Paragraph: Let $f(x)=\cos 2 x+x$ on $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$ Question: What is the greatest value of $f(x)$ ?

Consider the following for the two (02) items that follow: Let $ABC$ be a triangle right-angled at $B$ and $AB + AC = 3$ units. What is $\angle A$ equal to if the area of the triangle is maximum?

Paragraph: Let $f(x)=\cos 2 x+x$ on $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$ Question: What is the least value of $f(x)$ ?

What is the slope of the tangent to the curve $y=\sin ^{-1}$ $\left(\sin ^2 x\right)$ at $x=0$ ?