Mathematics - Application of Derivative Question with Solution | TestHub

(A) If $y=10-|x-10|,x\in [-1,3]:$ $-11\le x-10\le -7$ $\Rightarrow 7\le |x-10|\le 11$$\Rightarrow y\in [-1,3],\therefore$ maximum value of $y=3$. (B) Equation of tangent at $P$ is $ty=x+t^2$, it intersects the line $x=0$ at $Q,\therefore$ coordinates of $Q$ are $(0,t)$. Area of $△PQS=\frac{1}{2}\left|t+t^3\right|$ and $\frac{dA}{dt}=\frac{1}{2}\left(3t^2+1\right)>0$ $\forall t\in [0,2],\therefore$ area is maximum for $t=2$, maximum area $=5$. (C) As $a+b=1,$ $\sqrt{\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)}$ $=\sqrt{1+\frac{2}{ab}}$ and $\sqrt{ab}\le \frac{a+b}{2}=\frac{1}{2}\Rightarrow \frac{1}{ab}\ge 4,\therefore$ minimum value $=\sqrt{1+8}=3$. (D) Let $y=\frac{x+2}{2x^2+3x+6}$ $\Rightarrow 2y{x}^2$ $+x(3y-1)+6y-2=0$ $\Rightarrow D\ge 0$ $\Rightarrow (3y-1{)}^2-8y(6y-2)\ge 0$ $\Rightarrow y\in \left[-\frac{1}{13},\frac{1}{3}\right]$ .