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) = x² -4x + 3 on [1, 3]

✓

Answer

The given function is f(x) = x² -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) = 1² - 4 × 1 + 3 = 0
f(4) = 4² - 4 × 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
⇒ 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 "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

Two tailors, A and B earn Rs. 15 and Rs. 20 per day respectively. A can stitch 6 shirts and 4 pants while B can stitch 10 shirts and 4 pants per day. How many days shall each work if it is desired to produce (at least) 60 shirts and 32 pants at a minimum labour cost?

Solve the following initial value problems:
$(\text{xy}-\text{y}^2)\text{dx}-\text{x}^2\text{dy}=0,\text{y}(1)=1$

For each of the differential equation in find the particular solution satisfying the given condition:

$(\text{x}+\text{y})\ \text{dy}+(\text{x}-\text{y})\ \text{dx}=0;\ \text{y}=1\ \text{when}\ \text{x}=1$

Find the area bounded by the lines y = 4x + 5, y = 5 - x and 4y = x + 5.

$\text{Let}\ \vec{\text{a}}=4\hat{\text{i}}+5\hat{\text{j}}-\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}}$ and $\vec{\text{c}}=3\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}$ Find a vector $\vec{\text{d}}$ which is perpendicular to both $\vec{\text{c}}\ \text{and }\vec{\text{b}}\ \text{and}\ \vec{\text{d}}\cdot\vec{\text{a}}=21.$

Evaluate the following integrals:
$\int\limits^{\text{a}}_{-\text{a}}\frac{1}{1+\text{a}^{\text{x}}}\text{ dx},\text{ a}>0$

Find the equation of the plane mid-parallel to the planes 2x - 2y + z + 3 = 0 and 2x - 2y + z + 9 = 0

In each of the form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y = e^x (acos x + bsin x)

Find the distance of the point (- 1, - 5, - 10), from the point of intersection of the line
$\overrightarrow{\text{r}}=(2\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k})}+\lambda(3\hat{\text{i}}+4\hat{\text{j}}+2\hat{\text{k})}\text{ and the plane}\overrightarrow{\text{r}}\cdot(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k})}=5.$

Integrate the function in Exercise:
$(\text{x}^2+1)\text{log}\ \text{x}$