Download App

Mean Value Theorems — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSMean Value Theorems5 Marks
Question
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.
$f(x) = x(x + 4)^2$ on $[0,4]$
✓

Answer

Here, $f(x) = x(x + 4)^2$
$\Rightarrow f(x) = x(x^2+ 16 + 8x)$
$\Rightarrow f(x) = x^3 + 8x^2+ 16x$
Since $f(x)$ is a polynomial function which is everywhere continuous and differentiable.
Therefore, $f(x)$ is continuous on $[0, 4]$ and derivable on $(0, 4)$
Thus, both the conditions of Lagrange's theorem is satisfied.
Consequently, there exists some $\text{c}\in(0,4)$ such that
$\text{f}'(\text{c})=\frac{\text{f}(4)-\text{f}(0)}{4-0}=\frac{\text{f}(4)-\text{f}(0)}{4}$
Now, $f(x) = x^3 + 8x^2+ 16x$
$\Rightarrow f(x) = 3x^2+ 16x + 16,$
$\Rightarrow f(4) = 64+ 128 + 64 = 256,$
$⇒ f(0) = 0$
$\therefore\ \text{f}'(\text{x})=\frac{\text{f}(4)-\text{f}(0)}{4-0}$
$\Rightarrow3\text{x}^2+16\text{x}+16=\frac{256}{4}$
$\Rightarrow3\text{x}^2+16\text{x}-48=0$
$\Rightarrow\text{x}=-\frac{4}{3}\big(2+\sqrt{13}\big),\frac{4}{3}\big(\sqrt{13}-2\big)$
Thus, $\text{c}=\frac{-8+4\sqrt{13}}{3}\in(0,4)$ such that $\text{f}'(\text{c})=\frac{\text{f}(4)-\text{f}(0)}{4-0}$
Hence, Lagrange's mean value theorem is verified.

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 Mean Value Theorems→Mean Value Theorems — 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 derivative of sin (3x + 2) w.r.t. x from first principle.
→
Solve the following systems of linear equations by cramer's rule:
x - 2y = 4,
-3x + 5y = -7
→
A diet of two foods $F_1$ and $F_2$ contains nutrients thiamine, phosphorous and iron.
The amount of each nutrient in each of the food $($in milligrams per $25\ gms)$ is given in the following table:
Nutrients Food $F_1$ $F_2$
Thiamine $0.25$ $0.10$
Phosphorous $0.75$ $1.50$
Iron $1.60$ $0.80$
The minimum requirement of the nutrients in the diet are $1.00\ mg$ of thiamine, $7.50\ mg$ of phosphorous and $10.00\ mg$ of iron.
The cost of $F_1$ is $20$ paise per $25$ gms while the cost of $F_2$ is $15$ paise per $25$ gms.
Find the minimum cost of diet.
→
Arun and Tarun appeared for an interview for two vacancies. The probability of Arun's selection is $\frac{1}{4}$ and that to Tarun's rejection is $\frac{2}{3}$. Find the probability that at least one of them will be selected.
→
Evaluate the following integrals:
$\int\limits_{0}^{\frac{\pi}{4}}\frac{\sin\text{x}+\cos\text{x}}{3+\sin2\text{x}}\text{ dx}$
→
A bag A contains 4 black and 6 red balls and bag B contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B. If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black.
→
Show that the relation $R$ in the set $A=\{1,2,3,4,5\}$ given by $R=\{(a, b):|a-b|$ is divisible by 2$\}$ is an equivalence relation. Write all the equivalence classes of $R$.
→
If O is the origin and the coordinates of A are (a, b, c) Find the direction cosines of OA and the equation of the plane through A at right angles to OA.
→
A company sells two different products, $A$ and $B.$ The two products are produced in a common production process, which has a total capacity of $500$ man$-$hours. It takes $5$ hours to produce a unit of $A$ and $3$ hours to produce a unit of $B.$ The market has been surveyed and company officials feel that the maximum number of unit of $A$ that can be sold is $70$ and that for $B$ is $125.$ If the profit is $Rs. 20$ per unit for the product $A$ and $Rs. 15$ per unit for the product $B,$ how many units of each product should be sold to maximize profit?
→
Five bad oranges are accidently mixed with 20 good ones. If four oranges are drawn one by one successively with replacement, then find the probability distribution of number of bad oranges drawn. Hence find the mean and variance of the distribution.
→