Application of Derivatives

Consider the function $f(x)=0.75 x^4-x^3-9 x^2+7$ Consider the following statements: The function attains local minima at $x=-2$ and $x=3$. The function increases in the interval $(-2,0)$. Which of the above statements…

Paragraph : Consider an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Question : What is the area of the greatest rectangle that can be inscribed in the ellipse?

Paragraph: A curve y = $m e^{m x}$ where m > 0 intersects y-axis at a point P. Question: What is the slope of the curve at the point of intersection $P$ ?

What is the maximum area of a triangle that can be inscribed in a circle of radius a?

Consider the following statements : The derivative where the function attains maxima or minima must be zero. If a function is differentiable at a point, then it must be continuous at that point. Which of the above…

Consider the following statements: $y=\frac{e^x+e^{-x}}{2}$ is an increasing function on $[0, \infty)$. $y=\frac{e^x-e^{-x}}{2}$ is an increasing function on $(-\infty, \infty)$. Which of the above statements is/are…

Consider the following statements: $f(x)=\ln x$ is an increasing function on $(0, \infty)$. $\quad f(x)=e^x-x(\ln x)$ is an increasing function on $(1, \infty)$. Which of the above statements is/are correct?

Consider the following statements in respect of the function $f(x)=\sin x$ : $f(x)$ increases in the interval $(0, \pi)$. $f(x)$ decreases in the interval $\left(\frac{5 \pi}{2}, 3 \pi\right)$. Which of the above…

The function $f(x)=x^2-4 x, x \in[0,4]$ attains minimum value at

A given quantity of metal is to be cast into a half cylinder (i,e, with a rectangular base and semicircular ends). If the total surface area is to be minimum, then the ratio of the height of the half cylinder to the…

In which one of the following intervals is the function $f(x)=x^2-5 x+6$ decreasing?

Consider the following statements : The function $f(x)=x^2+2 \cos x$ is increasing in the interval $(0, \pi)$ The function $f(x)=\ln \left(\sqrt{1+x^2}-x\right)$ is decreasing in the interval $(-\infty, \infty)$ Which…