If the value of c prescribed bye Lagrange's mean value theorem fo… - Vidyadip

Question

If the value of c prescribed bye Lagrange's mean value theorem for the function
$\text{f}(\text{x})=\sqrt{\text{x}^2-4}$ defined on [2, 3].

✓

Answer

Here,
$\text{f}(\text{x})=\sqrt{\text{x}^2-4}$ defined on [2, 3].
We have to find c prescribed by Lagrange's mean value theorem, so
$\text{f}'(\text{c})=\frac{\text{f}(\text{b})-\text{f}(\text{a})}{\text{b}-\text{a}}$
$\Rightarrow\frac{2\text{c}}{2\sqrt{\text{c}^2-4}}=\frac{(\sqrt{9-4})-(\sqrt{4-4})}{3-2}$
$\Rightarrow\frac{\text{c}}{\sqrt{\text{c}^2-4}}=\frac{\sqrt5-0}{1}$
$\Rightarrow\frac{\text{c}}{\sqrt{\text{c}^2-4}}=\sqrt5$
Squaring both sides,
⇒ c² = (c²- 4)5
⇒ 5c² - c² = 20
⇒ 4c² = 20
⇒ c² = 5
$\Rightarrow\text{c}=\pm\sqrt5$
but $\text{c}=\sqrt5\text{ as }\sqrt5\in(2,3).$

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 "4 Marks" from Mean Value Theorems→Mean Value Theorems — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find which of the function:
$\text{f(x)}=\begin{cases}|\text{x}|\cos\frac{1}{\text{x}},&\text{if x}\neq0\\0,&\text{if x}=0\end{cases}$
at x = 0

Find the vector equation of the plane passing through three points with position vectors $\hat{\text{i}} + \hat{\text{j}} - 2\hat{\text{k}} , 2\hat{\text{i}} - \hat{\text{j}} + \hat{\text{k}}\text{ and } \hat{\text{i}} + 2\hat{\text{j}} + \hat{\text{k}}.$Also find the coordinates of the point of intersection of this plane and the line $\overrightarrow{\text{r}} = 3\hat{\text{i}} - \hat{\text{j}} - \hat{\text{k}} + \lambda(2\hat{\text{i}}- 2 \hat{\text{j}} + \hat{\text{k}}).$

A producer has 30 and 17 units of labour and capital respectively which he can use to produce two type of goods x and y. To produce one unit of x, 2 units of labour and 3 units of capital are required. Similarly, 3 units of labour and 1 unit of capital is required to produce one unit of y. If x and y are priced at Rs. 100 and Rs. 120 per unit respectively, how should be producer use his resources to maximize the total revenue? Solve the problem graphically.

Solve the following differential equation:
$\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}-\text{x}\cos^2\Big(\frac{\text{y}}{\text{x}}\Big)$

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

A trust invested some money in two type of bonds. The first bond pays 10% interest and bond pays 12% interest. The trust received 2,800 as interest. However, if trust had interchanged money in bonds, they would have got 100 less as interest. Using matrix method, find the amount invested by the trust. Which value is reflected in this question?

A dealer in rural area wishes to purchase a number of sewing machines. He has only ₹ 5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine costs him ₹ 360 and a manually operated sewing machine ₹240. He can sell the sewing machine at a profit of ₹ 22 and a manually operated sewing machine at a profit of ₹ 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as an LPP and solve it graphically.

The rate of growth of a population is proportional to the number present. If the population of a city doubled in the past 25 year, and the present population is 100000, when will the city have a population of 500000?

Verify Rolle's theorem for the following function on the indicated intervals

f(x) = (x - 1) (x - 2)² on [1, 2]

A publisher sells a hard cover edition of a text book for Rs. 72.00 and paperback edition of the same ext for Rs. 40.00. Costs to the publisher are Rs. 56.00 and Rs. 28.00 per book respectively in addition to weekly costs of Rs. 9600.00. Both types require 5 minutes of printing time, although hardcover requires 10 minutes binding time and the paperback requires only 2 minutes. Both the printing and binding operations have 4,800 minutes available each week. How many of each type of book should be produced in order to maximize profit?