Find two positive numbers whose sum is 16 and the sum of whose cu… - Vidyadip

Question

Find two positive numbers whose sum is $16$ and the sum of whose cubes is minimum.

✓

Answer

Let one number be $x$. Then, the other number is $(16 - x)$.
Let S(x) be the sum of these number. Then,
$S(x) = x^3 + (16 - x)^3$
$\Rightarrow S\ ' (x) = 3x^2 - 3(16 - x)^2$
$\Rightarrow S\ '' (x) = 6x + 6(16 - x)$
Now $, S\ ' (x) = 0$
$\Rightarrow 3x^2 - 3(16 - x)^2 = 0$
$\Rightarrow x^2 - (16 - x)^2 = 0$
$\Rightarrow x^2 - 256 - x^2 + 32x = 0$
$\Rightarrow x = 8$
Now $, S\ '' (8) = 6(8) + 6(16 - 8)$
$= 48 + 48 = 96 > 0$
Then, by second derivative test,$ x = 8$ is the point of local minima of $S.$
Therefore, the sum of the cubes of the numbers is the minimum when the numbers are $8$ and $16 - 8 = 8$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Application of Derivatives→Application of Derivatives — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the shortest distance between the following given lines $l_1$ and $l_2$
$\vec{r}=\hat{i}+2 \hat{j}-4 \hat{k}+\lambda(2 \hat{i}+3 \hat{j}+6 \hat{k})$
$\text { and } \vec{r}=3 \hat{i}+3 \hat{j}-5 \hat{k}+\lambda(2 \hat{i}+3 \hat{j}+6 \hat{k})$

An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed to be known:
P(A fails) = 0.2
P(B fails alone) = 0.15
P(A and B fail) = 0.15
Evaluate the following probabilities:

P(A fails|B has failed)
P(A fails alone)

Find the angle between the plane:
2x - y + z = 4 and x + y + 2z = 3

$\int\frac{2\text{x}+3}{(\text{x}-1)^2}\text{dx}$

If $\int\Big(\frac{\text{x}-1}{\text{x}^2}\Big)\text{e}^{\text{x}}\text{ dx}=\text{f(x)}\text{e}^{\text{x}}+\text{C},$ then write the value of f(x).

Solve system of linear equations, using matrix method.

3x + 4y - 5z = -5

2x - y + 3z = 12

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.$\text{f}(\text{x})=\sin\text{x}-\sin2\text{x}-\text{x}\text{ on }[0,\pi]$

Find the value of $\tan^{-1}\Big(\tan\frac{5\pi}{6}\Big)+\cos^{-1}\Big(\cos\frac{13\pi}{6}\Big).$

If $f{\text{x}}=\frac{(4\text{x}+3)}{(6\text{x}-4)},\ \text{x}\neq\frac{2}{3},$ show that fof(x) = x, for all $\text{x}\neq\frac{2}{3}.$ What is the inverse of f?

An urn contains $10$ white and $3$ black balls. Another urn contains $3$ white and $5$ black balls. Two are drawn from first urn and put into the second urn and then a ball is drawn from the latter. Find the probability that its is a white ball.