Applications of Derivatives

If $y=y(x)$ is the solution of the differential equation $x\dfrac{dy}{dx}+2y=xe^x$, $y(1)=0$ then the local maximum value of the function $z(x)=x^2y(x)-e^x$, $x\in\mathbb R$ is :

The number of distinct real roots of $x^4-4x+1=0$ is :

If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $f(x)=\displaystyle\int_0^{x^2}\frac{t^2-5t+4}{2+e^t}dt$, then the ordered pair $(m,n)$ is equal to

The number of real solutions of $x^7+5x^3+3x+1=0$ is equal to ______.

Let $l$ be a line which is normal to the curve $y=2x^2+x+2$ at a point P on the curve. If the point $Q(6,4)$ lies on the line $l$ and O is origin, then the area of the triangle OPQ is equal to ______.

A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is :

The number of solutions of the equation $2\theta-\cos^2\theta+\sqrt2=0$ in $\mathbf R$ is equal to ______.

Let $f:\mathbb R\to\mathbb R$ be a function defined by $f(x)=(x-3)^{n_1}(x-5)^{n_2}$, $n_1,n_2\in\mathbb N$. Then, which of the following is NOT true?

Let $f$ and $g$ be twice differentiable even functions on $(-2,2)$ such that $f\left(\frac14\right)=0$, $f\left(\frac12\right)=0$, $f(1)=1$ and $g\left(\frac34\right)=0$, $g(1)=2$. Then, the minimum number of solutions of $f(x)g''(x)+f'(x)g'(x)=0$ in $(-2,2)$ is equal to ______.

If $xy^4$ attains maximum value at the point $(x,y)$ on the line passing through the points $(50+\alpha,0)$ and $(0,50+\alpha)$, $\alpha>0$, then $(x,y)$ also lies on the line :

Let $f(x)=4x^3-11x^2+8x-5$, $x\in\mathbf R$. Then $f$ :

If the absolute maximum value of the function $f(x)=\left(x^2-2x+7\right)e^{\left(4x^3-12x^2-180x+31\right)}$ in the interval $[-3,0]$ is $f(\alpha)$, then :

The curve $y(x)=ax^3+bx^2+cx+5$ touches the $x$-axis at the point $P(-2,0)$ and cuts the $y$-axis at the point Q, where $y'$ is equal to 3. Then the local maximum value of $y(x)$ is :

If the maximum value of the term independent of $t$ in the expansion of $\left(t^2x^{\frac15}+\dfrac{(1-x)^{\frac1{10}}}t\right)^{15}$, $x\ge0$, is K, then 8 K is equal to ______.

The sum of the maximum and minimum values of the function $f(x)=|5x-7|+\left[x^2+2x\right]$ in the interval $\left[\dfrac54,2\right]$, where $[t]$ is the greatest integer $\le t$, is ______.

Let the area enclosed by the $x$-axis, and the tangent and normal drawn to the curve $4x^3-3xy^2+6x^2-5xy-8y^2+9x+14=0$ at the point $(-2,3)$ be A. Then 8A is equal to ______.

Let $f(x)=\begin{cases}x^3-x^2+10x-7, & x\le1\\-2x+\log_2\left(b^2-4\right), & x>1\end{cases}$. Then the set of all values of b, for which $f(x)$ has maximum value at $x=1$, is :

If $\dfrac{dy}{dx}+2y\tan x=\sin x$, $0<x<\dfrac\pi2$ and $y\left(\dfrac\pi3\right)=0$, then the maximum value of $y(x)$ is :

Let the function $f(x)=2x^2-\log_ex$, $x>0$, be decreasing in $(0,a)$ and increasing in $(a,4)$. A tangent to the parabola $y^2=4ax$ at a point P on it passes through the point $(8a,8a-1)$ but does not pass through the point $\left(-\dfrac1a,0\right)$. If the equation of the normal at P is $\dfrac x\alpha+\dfrac y\beta=1$, then $\alpha+\beta$ is equal to ______.

Let P and Q be any points on the curves $(x-1)^2+(y+1)^2=1$ and $y=x^2$, respectively. The distance between P and Q is minimum for some value of the abscissa of P in the interval