Download App

Mean Value Theorems — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMean Value Theorems4 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 - 1) on [1, 2]
✓

Answer

We have
f(x) = x(x - 1) which can be rewritten as f(x) = x2 - x
Since a polynomial function is everywhere continuous and differentiable.
Therefore, f(x) is continuous on [1, 2] and differentiable on (1, 2).
Thus, both conditions of Lagrange's mean value theorem is satisfied.
So, there must exist at least one real number $\text{c}\in(1,2)$ such that
$\text{f}'(\text{c})=\frac{\text{f}(2)-\text{f}(1)}{2-1}$
Now, f(x) = x2 - x
⇒ f'(x) = 2x - 1,
⇒ f(2) = 2, f(1) = 0
$\therefore\ \text{f}'(\text{x})=\frac{\text{f}(2)-\text{f}(1)}{2-1}$
$\Rightarrow2\text{x}-1=\frac{2-0}{2-1}$
$\Rightarrow2\text{x}-1-2=0$
$\Rightarrow2\text{x}=3$
$\Rightarrow\text{x}=\frac{3}{2}$
Thus, $\text{c}=\frac{3}{2}\in(1,2)$ such that $\text{f}'(\text{c})=\frac{\text{f}(2)-\text{f}(1)}{2-1}$
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 "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

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}} = \tan(\text{x}+\text{y})$
→
Show that the following systems of linear equations has infinite number of solutions and solve:
2x + y - 2z = 0,
x - 2y + z = -2,
5x - 5y + z = -2
→
Using integration find the area of the region $\big\{\text{(x, y) : x}^{2} + \text{y}^{2} \leq 2\text{ax,y}^{2}\geq \text{ax, x, y}\geq 0.\big\}$
→
Prove that $\begin{vmatrix}a^2&bc&ac+c^2\\a^2+ab&b^2&ac\\ab&b^2+bc&c^2\end{vmatrix}=4a^2b^2c^2$
→
Evaluate the following intregals:
$\int\frac{\text{x}^2+1}{(\text{x}-2)^2(\text{x}+3)}\ \text{dx}$
→
If A = diag(a, b, c), show that An = diag(an, bn, cn) for all positive integer n.
→
Find the intervals in which the following functions are increasing or decreasing.
f(x) = 2x3 - 24x + 107
→
Evaluvate the following intregals:
$\int\frac{1}{1-\cot\text{x}}\text{ dx}$
→
If $\sin^{-1}\text{x}+\sin^{-1}\text{y}=\frac{\pi}{3}$ and $\cos^{-1}\text{x}-\cos^{-1}\text{y}=\frac{\pi}{6},$ find the values of x and y.
→
By computing the shortest distance determine whether the following pairs of lines intersect or not:
$\frac{\text{x}-1}{2}=\frac{\text{y}+1}{3}=\text{z}$ and $\frac{\text{x}+1}{5}=\frac{\text{y}-2}{1};\text{z}=2$
→