Download App

Mean Value Theorems — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMean Value Theorems5 Marks
Question
If $\text{f}:[-5,5]\rightarrow\text{R}$ is differentiable and if f'(x) does not vanish anywhere, then prove that $\text{f}(-5)\pm\text{f}(5).$
✓

Answer

It is given that $\text{f}:[-5,5]\rightarrow\text{R}$ is a differentiable function.
Since every differentiable function is continuous function, we obtain
  1. f is continuous on [-5, 5].
  2. f is differentiable on (-5, 5).
Therefore, by the Mean Value Theorem, there exists $\text{c}\in(-5,5)$ such that

$\text{f}'(\text{c})=\frac{\text{f}(5)-\text{f}(-5)}{5-(-5)}$

$\Rightarrow10\text{f}'(\text{c})=\text{f}(5)-f(-5)$

It is also given that f'(x) does not vanish anywhere.

$\therefore\ \text{f}'(\text{c})\neq0$

$\Rightarrow10\text{f}'(\text{c})\neq0$

$\Rightarrow\text{f}(5)-\text{f}(-5)\neq0$

$\Rightarrow\text{f}(5)\neq\text{f}(-5)$

Hence, proved.

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 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

A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is $10 m.$ Find the dimensions of the window to admit maximum light through the whole opening
→
Differentiate $\sin^{-1}\Big(2\text{x}\sqrt{1-\text{x}^2}\Big)$ with respect to $\sec^{-1}\Big(\frac{1}{\sqrt{1+\text{x}^2}}\Big),$ if:
$\text{x}\in\Big(0,\frac{1}{\sqrt{2}}\Big)$
→
Two schools P and Q want to award their selected students on the values of Tolerance, Kindness and Leadership. The school P wants to award ₹ x each, ₹ y each and ₹ z each for the three respective values to 3, 2 and 1 students respectively with a total award money of ₹ 2,200. School Q wants to spend ₹ 3,100 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school P). If the total amount of award for one prize on each values is ₹ 1,200, using matrices, find the award money for each value.
Apart from these three values, suggest one more value which should be considered for award.
→
If X is the number of tails in three tosses of a coin, determine the standard deviation of X.
→
Verify Rolle's theorem for the following function on the indicated intervals $f(x) = (x - 1) (x - 2)^2$ on $[1, 2]$
→
Five forces $\overrightarrow{\text{AB}},\ \overrightarrow{\text{AC}},\ \overrightarrow{\text{AD}},\ \overrightarrow{\text{AE}}\text{ and }\overrightarrow{\text{AF}}$ act at the vertex of a regular hexagon ABCDEF. Prove that the resultant is:$6\ \overrightarrow{\text{AO}}$, where o is the center of hexagon.
→
Show that the points $(1, 1, 1)$ and $(-3, 0, 1)$ are equidistant from the plane $3x + 4y - 12z + 13 = 0$.
→
If $\text{x}=3\sin\text{t}-\sin3\text{t},$ $\text{y}=3\cos-\cos3\text{t}$ find $\frac{\text{dy}}{\text{dx}}$ at $\text{t}=\frac{\pi}{3}.$
→
Show that the matrix $\text{A}=\begin{bmatrix}5&3\\12&7\end{bmatrix}$ is root of the equation $A^2 - 12A - I = 0$.
→
Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
$(x + y)^2 = 2axy$
→