1. Tardigrade
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  4. An open topped box is to be constructed by removing equal squares from each corner of a 3 m by 8 m rectangular sheet of aluminium and folding up the sides. The largest volume of such a box is

Q. An open topped box is to be constructed by removing equal squares from each corner of a $3 m$ by $8 m$ rectangular sheet of aluminium and folding up the sides. The largest volume of such a box is

Application of Derivatives Report Error

Solution:

Let $x m$ be the length of a side of the removed squares. Then, the height of the box is $x$, length is $8-2 x$ and breadth is $3-2 x$. If $V(x)$ is the volume of the box, then
image
Therefore, $V(x) =x(3-2 x)(8-2 x)$
$ =4 x^3-22 x^2+24 x $
$\Therefore V^{\prime}(x) =12 x^2-44 x+24=4(x-3)(3 x-2) $
$ V^{\prime}(x) =24 x-44$
Now, $V^{\prime}(x)=0$ gives $x=3, \frac{2}{3}$. But $x \neq 3$
Thus, we have $x=\frac{2}{3}$. Now $V^{\prime \prime}\left(\frac{2}{3}\right)=24\left(\frac{2}{3}\right)-44=-28 < 0$.
Therefore, $x=\frac{2}{3}$ is the point of maxima, i.e., if we remove a square of side $\frac{2}{3} m$ from each corner of the sheet and make a box from the remaining sheet, then the volume of the box such obtained will be the largest and it is given by
$V\left(\frac{2}{3}\right)=4\left(\frac{2}{3}\right)^3-22\left(\frac{2}{3}\right)^2+24\left(\frac{2}{3}\right)=\frac{200}{27} m ^3$

Questions from Application of Derivatives

1. Let $ 𝑓 \left(x\right) = 𝑥 + log_{e} x − 𝑥 log_{e} 𝑥, 𝑥 ∈ \left(0,∞\right).$
• Column 1 contains information about zeros of $𝑓 (𝑥) , 𝑓'(𝑥)$ and $f ''(𝑥)$.
• Column 2 contains information about the limiting behavior of $𝑓(𝑥) , 𝑓 ' (𝑥)$ and $f ''(𝑥)$ at infinity.
• Column 3 contains information about increasing/decreasing nature of $𝑓(𝑥)$ and $f' (𝑥)$ .
Column 1 Column 2 Column 3
(I) $𝑓 (𝑥) = 0$ for some $x\in\left(1,e^{2}\right)$ (i) $\lim_{x\to\infty}f \left(x\right)=0$ (P) $𝑓$ is increasing in ($0, 1$)
(II) $𝑓(𝑥) = 0$ for some $x\in\left(1, e\right)$ (ii) $\lim_{x\to\infty}f \left(x\right)=-\infty$ (Q) $f$ is decreasing in ($𝑒, 𝑒^2$)
(III) $𝑓(𝑥) = 0$ for some $x\in\left(0, 1\right)$ (ii) $\lim_{x\to\infty}f' \left(x\right)=-\infty$ (R) $f'$ is increasing in ($0, 1$)
(IV) $f ''(𝑥) = 0$ for some $x\in\left(1, e\right)$ (ii) $\lim_{x\to\infty}f'' \left(x\right)=0$ (S) $f'$ is decreasing in ($𝑒, 𝑒^2$)

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