Question
Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\text{e}^{1-\text{x}^2}\text{ on }[-1,1]$
$\text{f}(\text{x})=\text{e}^{1-\text{x}^2}\text{ on }[-1,1]$
✓
Answer
Here,
$\text{f}(\text{x})=\text{e}^{1-\text{x}^2}\text{ on }[-1,1]$
We know that, exponantial function is continuous and differentiable everywhere. So, f(x) is continuous IS $[-1,1]$ and differentiable is $(-1,1).$ Now, $\text{f}(-1)=\text{e}^{1-1}=1$ $\text{f}(1)=\text{e}^{1-1}=1$ $\Rightarrow\text{f}(-1)=1$ So, Rolle's theorem is applicable, so there must exist a point $\text{c}\in(-1,1)$ such that f'(c) = 0. Now,$\text{f}(\text{x})=\text{e}^{1-\text{x}^2}$
$\text{f}'(\text{x})=\text{e}^{1-\text{x}^2}(-2\text{x})$ Now, $\text{f}'(\text{c})=0$ $-2\text{ce}^{1-\text{c}^2}=0$ $\Rightarrow\text{c}=0$ or $\text{e}^{1-\text{c}^2}=0$ $\Rightarrow\text{c}=0\in(-1,1)$ Hence, 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.
Explore more
Similar questions
Form the differential equation corresponding to $(\text{x}-\text{a})^2+(\text{y}-\text{b})^2=\text{r}^2$ by eliminating a and b.
→Determine if $\text{f(x)}=\begin{cases}\text{x}^2\sin\frac{1}{\text{x}},&\text{ x}\neq0\\0,&\text{x}=0\end{cases}$ is a continuous function?
→If $\text{y}=\text{x}^3\log\text{x},$ Prove that $\frac{\text{d}^4\text{y}}{\text{dx}^4}=\frac{6}{\text{x}}$
→If the median to the base of a triangle is perpendicular to the base, then triangle is isosceles.
If $\text{y}=\text{cosec}^{-1}\text{x},\text{x}>1$ prove that $\text{x}(\text{x}^2-1)\frac{\text{d}^2\text{y}}{\text{dx}^2}+(2\text{x}^2-1)\frac{\text{dy}}{\text{dx}}=0.$
→Find:
$\int \text{e}^{2\text{x}} \sin \text{(3x + 1)} \text{ dx}$
→$\int \text{e}^{2\text{x}} \sin \text{(3x + 1)} \text{ dx}$
Evaluate the following integrals:
$\int\sqrt{3-2\text{x}-2\text{x}^2}\text{dx}$
→$\int\sqrt{3-2\text{x}-2\text{x}^2}\text{dx}$
Find the cartesian form of the equations of the following planes.
$\vec{\text{r}}=(\hat{\text{i}}-\hat{\text{j}})+\text{s}(-\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}})+\text{t}(\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}})$
→$\vec{\text{r}}=(\hat{\text{i}}-\hat{\text{j}})+\text{s}(-\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}})+\text{t}(\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}})$
Find the dimensions of the rectangle of perimeter 36cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also find the maximum volume.
The feasible region for a LPP is shown in Evaluate Z = 4x + y at each of the corner points of this region. Find the minimum value of Z, if it exists.
