Applications of Derivatives JEE Main PYQs, 2020 to 2025

Let the function $f(x) = \frac{x}{3} + \frac{3}{x} + 3, x \neq 0$ be strictly increasing in $(-\infty, \alpha_1) \cup (\alpha_2, \infty)$ and strictly decreasing in $(\alpha_3, \alpha_4) \cup (\alpha_4, \alpha_5)$. Then $\sum\limits_{i=1}^{5} \alpha_i^2$ is equal to

Let f : ℝ $\to$ ℝ be a polynomial function of degree four having extreme values at x = 4 and x = 5. If $\lim\limits_{x \to 0} \frac{f(x)}{x^2} = 5$, then f(2) is equal to :

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a thrice differentiable function such that $f(0)=0, f(1)=1, f(2)=-1, f(3)=2$ and $f(4)=-2$. Then, the minimum number of zeros of $\left(3 f^{\prime} f^{\prime \prime}+f f^{\prime \prime \prime}\right)(x)$ is __________.

Let the quadratic curve passing through the point $(-1,0)$ and touching the line $y=x$ at $(1,1)$ be $y=f(x)$. Then the $x$-intercept of the normal to the curve at the point $(\alpha, \alpha+1)$ in the first quadrant is __________.

A wire of length $20 \mathrm{~m}$ is to be cut into two pieces. A piece of length $l_{1}$ is bent to make a square of area $A_{1}$ and the other piece of length $l_{2}$ is made into a circle of area $A_{2}$. If $2 A_{1}+3 A_{2}$ is minimum then $\left(\pi l_{1}\right): l_{2}$ is equal to :

The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is :