Verify Rolle's theorem of the following function on the indicated… - Vidyadip

Question

Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\text{x}^2-5\text{x}+4\text{ on }[1,4]$

✓

Answer

According to Rolle’s theorem, if f(x) is a real valued function defined on [a, b] such that it is continuous on [a, b], it is differentiable on (a, b) and f(a) = f(b), then there exists a real number $\text{c}\in(\text{a},\text{b})$ such that f(c)= 0.
Now, f(x) is defined for all $\text{x}\in[1,4].$
At each point of [1, 4], the limit of f(x) is equal to the value of the function.
Therefore, f(x) is continuous on [1, 4].
Also, f'(x) = 2x - 5 exists for all $\text{x}\in(1,4).$
So, f(x) is differentiable on (1, 4).
Also,
f(1) = f(4) = 0
Thus, all the three conditions of Rolle’s theorem are satisfied.
Now, we have to show that there exists $\text{c}\in(1,4)$ such that f'(c) = 0.
We have
f'(x) = 2x - 5
$\therefore$ f'(x) = 0
⇒ 2x - 5 = 0
$\Rightarrow\text{x}=\frac{5}{2}$
$\Big[\text{Since }\text{c}=\frac{5}{2}\in(1,4)\text{ such that }\text{f}'(\text{c})=0\Big]$
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.

Start Generating Free

Explore more

More "3 Marks Question" 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

Find the interval for which function $f(x)=x^3-$ $3 x^2-24 x+5$ is increasing.

Find the area of a triangle whose vertices are $A(1,1,1), B(1,2,3)$ and $C(2,3,3)$.

Check whether the relation $R$ on $R$ defined by $R = \{(a, b): a \leq b^3\}$ is reflexive, symmetric or transitive.

The following relation are defined on the set of real numbers.
aRb if $|\text{a}|\leq\text{b}$
Find whether these relation are reflexive, symmetric or transitive.

Evaluate the following integrals:
$\int\sqrt{\text{x}-\text{x}^2}\text{dx}$

Find the equation of the plane passing through the following point:
(-5, 0, -6), (-3, 10, -9) and (-2, 6, -6)

If $x ^{ x }+ y ^{ x }=1$, prove that $\frac{d y}{d x}=-\left\{\frac{x^x(1+\log x)+y^x \cdot \log y}{x \cdot y^{(x-1)}}\right\}$

Show that the function $f(x)$ defined by $f ( x )=\left\{\begin{array}{ll}\frac{\sin x}{x}+\cos x, & x>0 \\ 2, & x=0 \text { is continuous at } x =0 . \\ \frac{4(1-\sqrt{1-x})}{x}, & x<0\end{array}\right.$

The probability that it rains today is 0.4. If it rains today, the probability that it will rain tomorrow is 0.8. If it does not rain today, the probability that it will rain tomorrow is 0.7. If
P1: denotes the probability that it does not rain today.
P2: denotes the probability that it will not rain tomorrow, if it rains today.
P3: denotes the probability that it will rain tomorrow, if it does not rain today.
P4: denotes the probability that it will not rain tomorrow, if it does not rain today.
(i) Find the value of $P_1 \times P_4-P_2 \times P_3$
(ii) Calculate the probability of raining tomorrow.

Find the equation of the normal to $y = 2x^3 - x^2 + 3$ at $(1, 4).$