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 $f(x) = x^2 -4x + 3$ on $[1, 3]$

✓

Answer

The given function is $f(x) = x^2 -4x + 3$
$f,$ being a pollynomial function, is continuous in $[1, 4]$ and is differentiable in $(1, 4)$ whose derivative is $2x - 4.$
$f(1) = 1^2 - 4 \times 1 + 3 = 0$
$f(4) = 4^2 - 4 \times 4 + 3 = 3$
$\therefore\ \frac{\text{f}(\text{b})-\text{f}(\text{a})}{\text{b}-\text{a}}=\frac{\text{f}(4)-\text{f}(1)}{4-1}=\frac{3-(0)}{3}=\frac{3}{3}=1$
Mean Value Theorem states that there is a point $\text{c}\in(1,4)$ such that $f'(c) = 1$
$f'(c) = 1$
$\Rightarrow 2c - 4 = 1$
$\Rightarrow\text{c}=\frac{5}{2},$ where $\text{c}=\frac{5}{2}\in(1,4)$
Hence, Mean Value Theorem is verified for the given function.

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 the line $\frac{\text{x}-3}{2}=\frac{\text{y}+2}{-1}=\frac{\text{z}+4}{3}$ lies in the plane $lx + my - z = 9,$ then find the value of $l^2 + m^2$.

Form the differential equation of the family of circle in the secound qudrant and touching the coordinate axes.

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\text{a}+\text{bx}}{\text{b}-\text{ax}}\Big)$

Integrate the function in Exercise:
$\text{e}^{2\text{x}}\sin\text{x}$

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.
$\text{f}(\text{x})=\text{x}+\frac{1}{\text{x}}\text{ on }[1,3]$

A company produces three product every day.Their production on a certain day is $45$ tons. It is found that the production of third product exceeds the production of first product by $8$ tons while the total production of first and third product is twice the production of second product. Determine the production level of each product using matrix method.

If A = $\begin{bmatrix}0&-\tan\frac{\alpha}{2}\\ \tan\frac{\alpha}{2}&0\end{bmatrix}$ and I is the identity matrix of order 2, show that I + A = ( I - A) $\begin{bmatrix}\cos\alpha&-\sin\alpha\\ \sin{\alpha}&\cos\alpha\end{bmatrix}.$

Prove that $\begin{vmatrix}a^2&bc&ac+c^2\\a^2+ab&b^2&ac\\ab&b^2+bc&c^2\end{vmatrix}=4a^2b^2c^2$

Find the distance of the point (1, -2, 3) from the plane x - y + z = 5 measured along a line parallel to $\frac{\text{x}}{2}=\frac{\text{y}}{3}=\frac{\text{z}}{-6}.$

$\int\sin\text{x}\sqrt{1-\cos2\text{x}}\text{ dx}$