Verify Rolle's theorem for the following function on the indicate… - Vidyadip

Question

Verify Rolle's theorem for the following function on the indicated intervals
$f(x) = x^2 - 8x + 12$ on $[2, 6]$

✓

Answer

Given: $f(x) = x^2 - 8x + 12$
We know that a polynomial function is everywhere derivable and hence continuous.
So, being a polynomial function $f(x)$ is continuous and derivable on $[2, 6].$
$f(2) = (2)^2 - 8(2) + 12 = 4 - 16 + 12 = 0$
$f(6) = (6)^2 - 8(6) + 12 = 36 - 48 + 12 = 0$
$\therefore f(2) = f(6) = 0$
Thus, all the conditions of rolle's theorem are satisfied.
Now, we have to show that there exist $\text{c}\in(2, 6)$ such that $f'(c) = 0$
We have
$f(x) = x^2 - 8x + 12$
$\Rightarrow f'(x) = 2x - 8$
$\therefore f'(x) = 0$
$\Rightarrow 2x - 8 = 0$
$\Rightarrow x = 4$
Thus, $\text{c}=4\in(2,6)$ such that $f'(c) = 0$

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

If $\cos\text{y}=\text{x}\cos(\text{a}+\text{y}),$ with $\cos\text{a}\neq\pm1,$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\cos^2(\text{a}+\text{y})}{\sin\text{a}}$

Evaluate the following integrals:
$\int\frac{\text{x}^2+9}{\text{x}^4+81}\ \text{dx}$

$\overrightarrow{A B}=3 \hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{C D}=-3 \hat{i}+2 \hat{j}+4 \hat{k}$ are two vectors. The position vectors of the points $A$ and $C$ are $6 \hat{i}+7 \hat{j}+4 \hat{k}$ and $-9 \hat{j}+2 \hat{k}$, respectively. Find the position vector of a point $P$ on the line $AB$ and a point $Q$ on the line $CD$ such that $\overrightarrow{P Q}$ is perpendicular to $\overrightarrow{A B}$ and $\overrightarrow{C D}$ both.

A manufacturer has employed 5 skilled men and 10 semi-skilled men and makes two models A and B of an article. The making of one item of model A requires 2 hours work by a skilled man and 2 hours work by a semi-skilled man. One item of model B requires 1 hour by a skilled man and 3 hours by a semi-skilled man. No man is expected to work more than 8 hours per day. The manufacturer’s profit on an item of model A is Rs. 15 and on an item of model B is Rs. 10. How many of items of each model should be made per day in order to maximize daily profit? Formulate the above LPP and solve it graphically and find the maximum profit.

$\text{If A}=\begin{bmatrix} 2 & 3 & 1 \\ 1 & 2 & 2 \\ -3 & 1 & -1 \end{bmatrix}$, find $A–1$ and hence solve the system of equations $2x + y – 3z = 13,$
$3x + 2y + z = 4, x + 2y – z = 8.$

An item is manufactured by three machines $A, B$ and $C$. Out of the total number of items manufactured during a specified period, $50\%$ are manufactured on $A, 30\%$ on $B$ and $20\%$ on $C. 2\%$ of the items produced on $A$ and $2\%$ of items produced on $B$ are defective, and $3\%$ of these produced on $C$ are defective. All the items are stored at one godown. One item is drawn at random and is found to be defective. What is the probability that it was manufactured on machine A?

Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
$\text{x}^{\frac{2}{3}}+\text{y}^{\frac{2}{3}}=\text{a}^{\frac{2}{3}}$

Show that the lines
$\vec{r}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+4 \hat{k})$ and $\vec{r}=(4 \hat{i}+\hat{j})+\mu(5 \hat{i}+2 \hat{j}+\hat{k})$
intersect. Also, find their point intersection.

The top of a ladder $6$ metres long is resting against a vertical wall on a level pavement, when the ladder begins to slide outwards. At the moment when the foot of the ladder is $4$ metres from the wall, it is sliding away from the wall at the rate of $0.5m/ \sec$. How fast is the top$-$sliding downwards at this instance?
How far is the foot from the wall when it and the top are moving at the same rate?

Using the method of integration, find the area of the region bounded by the lines 3x – 2y + 1 = 0, 2x + 3y – 21 = 0 and x – 5y + 9 = 0.