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+ 5 x + 6 $on the interval $[-3, -2]$

✓

Answer

Here, $f(x) = x^2+ 5 x + 6$ on $[-3, -2]$
f(x) is continuous is [-3, -2] and f(x) is differentiable is $(-3, -2)$ since it is a polynomial function.
Now,
$f(x) = x^2+ 5x + 6$
$f(-3) = (-3)^2+5(-3) + 6$
$= 9 - 15 + 6$
$f(-3) = 0 ....(i)$
$f(-2) = (-2)^2+ 5(-2) + 6$
$= 4 - 10 + 6$
$f(-2) = 20 ....(ii)$
From equation (i) and (ii),
$f(-3) = f(-2)$
So, Rolle's theorem is applicable is $[-3, -2]$, we have to show that
f'(c) = 0 as $\text{c}\in (-3,-2)$
Now,
$f(x) = x^2 + 5x + 6$
$f'(x) = 2x + 5$
$\Rightarrow f'(c) = 0$
$2c + 5 = 0$
$\text{c}=\frac{-5}{2}\in(-3,-2)$
So, 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 "5 Marks Questions" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Show that for $\text{a}\geq1,\text{ f(x)}=\sqrt{3}\sin\text{x}-\cos\text{x}-2\text{ax}+\text{b}$ is decreasing in R.

Evaluate: $\int\limits^{\pi}_{0} \frac{\text{x \sin x}}{1 + 3\cos^{2}\text{x}}\text{dx}$

Let $\vec{\text{u}},\vec{\text{v}}$ and $\vec{\text{w}}$ be vectors such $\vec{\text{u}}+\vec{\text{v}}+\vec{\text{w}}=\vec{0}.$ If $|\vec{\text{u}}|=3,|\vec{\text{v}}|=4$ and $|\vec{\text{w}}|=5,$ then find $\vec{\text{u}}.\vec{\text{v}}+\vec{\text{v}}.\vec{\text{w}}+\vec{\text{w}}.\vec{\text{u}}.$

Evaluate the following:
$\tan\Big\{2\tan^{-1}\frac{1}{5}-\frac{\pi}{4}\Big\}$

If $\text{u}=\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)$ and $\text{v}=\tan^{-1}\Big(\frac{2\text{x}}{1-\text{x}^2}\Big),$ where -1 < x < 1 then write the value of $\frac{\text{dy}}{\text{dx}}.$

Classify the following functions as injection, surjection or bijection:
$f : R \rightarrow R$, defined by$ f(x) = x^3 - x$

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}+\frac{\text{y}}{\text{x}}=\text{x}^3$

A box of constant volume $c$ is to be twice as long as it is wide. The material on the top and four sides cost three times as much per square metre as that in the bottom. What are the most economic dimensions?

Evaluate:
$\int\limits^{\pi/2}_{0} \frac{\cos^{2} \text{x dx}}{1 + 3\sin^{2}\text{x}}$

A bag contains 4 white and 5 black balls and another bag contains 3 white and 4 black balls. A ball is taken out from the first bag and without seeing its colour is put in the second bag. A ball is taken out from the latter. Find the probability that the ball drawn is white.