Download App

Maxima and Minima — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSMaxima and Minima5 Marks
Question
How should we choose two numbers, each greater than or equal to −2, whose sum so that the sum of the first and the cube of the second is minimum?
✓

Answer

Let the two number be x and y. Then,
$\text{x},\text{y}>-2 \ \text{and}\ \text{x}+\text{y}=\frac{1}{2}....(\text{i})$
Now, $\text{z}=\text{x}+\text{y}^{3}$
$\Rightarrow \text{z}=\text{x}+\Big(\frac{1}{2}-\text{x}\Big)^{3}$ [From eq. (i)]
$\Rightarrow\frac{\text{dz}}{\text{dx}}=1+3\Big(\frac{1}{2}-\text{x}\Big)^{2}$
For maximum or minimum values of z, We must have $\Rightarrow\frac{\text{dz}}{\text{dx}}=0$
$\Rightarrow1+3\Big(\frac{1}{2}-\text{x}\Big)^{2}=0$
$\Rightarrow\Big(\frac{1}{2}-\text{x}\Big)^{2}=\frac{1}{3}$
$\Rightarrow\Big(\frac{1}{2}-\text{x}\Big)=\pm\frac{1}{\sqrt{3}}$
$\Rightarrow\text{x}=\frac{1}{2}\pm\frac{1}{\sqrt{3}}$
$\frac{\text{d}^{2}\text{z}}{d\text{x}^{2}}=-36\text{x}$
At $\text{x}=\frac{1}{2}\pm\frac{1}{\sqrt{3}}$.

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 "5 Marks Questions" from Maxima and Minima→Maxima and Minima — 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 inverse of the following matrices and verify that $A^{-1} A = I_3.$
$\begin{bmatrix}2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2 \end{bmatrix}$
→
Prove that:
$\begin{vmatrix}\text{b}+\text{c}&\text{a}-\text{b}&\text{a}\\\text{c}+\text{a}&\text{b}-\text{c}&\text{b}\\\text{a}+\text{b}&\text{c}-\text{a}&\text{c}\end{vmatrix}=3\text{abc}-\text{a}^3-\text{b}^3-\text{c}^3$
→
Find $\frac{\text{dy}}{\text{dx}}$
$y = e^x + 10^x + x^x$
→
A dietician wishes to mix together two kinds of food $X$ and $Y$ in such a way that the mixture contains at least $10$ units of vitamin $A, 12$ units of vitamin $B$ and $8$ units of vitamin $C.$ The vitamin contents of one $\ kg$ food is given below:
Food
Vitamin $A$
Vitamin $B$
Vitamin $C$
$X$ $1$ $2$ $3$
$Y$ $2$ $2$ $1$
One $\ kg$ of food $X$ costs $Rs. 16$ and one $\ kg$ of food $Y$ costs $Rs. 20.$ Find the least cost of the mixture which will produce the required diet?
→
Determine the equation of the line passing through the points (1, 2, -4) and perpendicular to the lines $\frac{\text{x}-8}{8}=\frac{\text{y}+9}{-16}=\frac{\text{z}-10}{7}$ and $\frac{\text{x}-15}{3}=\frac{\text{y}-29}{8}=\frac{\text{z}-5}{-5}.$
→
Evaluate the following integrals as limit of sum:
$\int\limits^4_{1}\big(\text{x}^2-\text{x}\big)\text{dx}$
→
Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\sin\text{x}\sin2\text{x}\sin3\text{x}\sin4\text{x}$
→
Solve the following initial value problems:
$\Big\{\text{x}\sin^2\Big(\frac{\text{y}}{\text{x}}\Big)-\text{y}\Big\}\text{dx + x dy}=0,\text{y}(1)=\frac{\pi}4$
→
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
→
If A and B are two events such that $\text{P}(\text{A})=\frac{1}{3},\text{P(B)}=\frac{1}{5}$ and $\text{P}(\text{A}\cup\text{B})=\frac{11}{30}$ find $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)$ and $\text{P}\Big(\frac{\text{B}}{\text{A}}\Big).$
→