Verify Rolle's theorem for the following function: f(x)=x^2-4 x+1… - Vidyadip

Question

Verify Rolle's theorem for the following function: $f(x)=x^2-4 x+10$ on $[0,4]$

✓

Answer

Since $f (x)$ is a polynomial,
$(i)$ It is continuous on $[0, 4]$
$(ii)$ It is differentiable on $(0, 4)$
$(iii) \ f (0) = 10, f (4) = 16 - 16 + 10 = 10$
$\therefore f(0) = f(4) = 10$
Thus all the conditions on Rolle’s theorem are satisfied
The derivative of $f (x)$ should vanish for at least one point $c$ in $(0, 4)$. To obtain the value of $c,$ we proceed as follows
$f(x) = x^2 - 4x + 10$
$f'(x) = 2x - 4 = 2(x - 2)$
$\therefore f'(x) = 0 \Rightarrow (x - 2) = 0$
$\therefore x= 2$
$\therefore ∃c = 2$ in $(0,4)$
We know that $2 \in (0, 4)$
Thus Rolle’s 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 "Solve the Following Question.(4 Marks)" from Applications of Derivatives→Applications of Derivatives — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the shortest distance between lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-2}{3}=\frac{y-4}{4}=\frac{z-5}{5}$

Prove that three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are coplanar if and only if there exists non-zero linear combination $x \vec{a}+y \vec{b}+z \vec{c}=\overrightarrow{0}$.

Differentiate the following w.r.t. x:

$\cos ^{-1}\left(\frac{\sqrt{1+x}-\sqrt{1-x}}{2}\right)$

A point source of light is hung 30 feet directly above a straight horizontal path on which a man of 6 feet in height is walking. How fast will the man's shadow lengthen and how fast the tip of shadow is move when he is walking away from the light at the rate of $100 ft / min$

Examine whether the statement pattern

[p → (∼q ˅ r)] ↔ ∼[p → (q → r)] is a tautology, contradiction or contingency.

Show that the height of a right circular cylinder of greatest volume that can be inscribed in a right circular cone is one-third of that of the cone.

Parametric form of the equation of the plane is $\bar{r}=(2 \bar{i}+4 \bar{k})-\lambda \bar{i}+\mu(\bar{i}+2 \bar{j}-3 \bar{k}) \lambda$ and $\mu$ are parameters. Find normal to the plane and hence equation of the plane in normal form. Write its Cartestan form.

Find the area of the region bounded by the curves. $y^2=4 x$ and $4 x^2+4 y^2=9$ with $x \geq 0$.

Find the component form of if a if

It lies in XZ plane and makes $45^{\circ}$ with positive $Z$-axis and $|\bar{a}|=10$

The function $f(x)=x(x+3) e^{-\frac{x}{2}}$ satisfies all the conditions of Rolle's theorem on $[-3,0]$. Find the value of $c$ such that $f^{\prime}(c)=0$.