Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
It is given that the Rolle's theorem holds for the function $f(x) = x^3 + bx^2 + cx, \text{x}\in[1,2]$ at the point $\text{x}=\frac{4}{3},$ the values of b and c.
✓

Answer

So, $f(1) = f(2)$
$\Rightarrow (1)^3 + b(1)^2 + c(1) = (2)^3 + b(2)^2 + c(2)$
$\Rightarrow 1 + b + c = 8 + 4b + 2c$
$\Rightarrow 3b + c + 7 = 0$ ....(i)
And $\text{f}'\Big(\frac{4}{3}\Big)=0$
$\Rightarrow3\Big(\frac{4}{3}\Big)^2+2\text{b}\Big(\frac{4}{3}\Big)+\text{c}=0$ [As, $f'(x) = 3x^2 + 2bx + c$]
$\Rightarrow\frac{16}{3}+\frac{8\text{b}}{3}+\text{c}=0$
$\Rightarrow8\text{b}+3\text{c}+16=0\ ....(\text{ii})$
(ii) - (i) × 3, we get
$8b - 9b + 16 - 21 = 0$
$\Rightarrow -b - 5 = 0$
$\Rightarrow b = -5$
Substituting $b = -5$ in (i), we get
$3(-5) + c + 7 = 0$
$\Rightarrow -15 + c + 7 = 0$
$\Rightarrow c = 8$

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

If $\text{x}^\text{m}.\text{y}^\text{n}=(\text{x}+\text{y})^{\text{m}+\text{n}},$ prove that:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}$
→
A bag contains 25 tickets, numbered from 1 to 25. A ticket is drawn and then another ticket is drawn without replacement. Find the probability that both tickets will show even numbers.
→
Find the absolute maximum and the absolute minimum value of the following functions in the given intervals:
$\text{f}(\text{x})=(\text{x}-2)\sqrt{\text{x}-1}\ \text{in}\ [1,9]$
→
Write the vector equation of the following lines and hence determine the distance between them $\frac{\text{x}-1}{2}=\frac{\text{y}-2}{3}=\frac{\text{z}+4}{6}$ and $\frac{\text{x}-3}{4}=\frac{\text{y}-3}{6}=\frac{\text{z}+5}{12}$
→
Solve the following systems of linear equations by cramer's rule:
3x + y = 19,
3x - y = 23
→
Evaluate the definite integral in Exercise:
$\int\limits^{\frac{\pi}{2}}_{0}\frac{\cos^{2}\text{dx}}{\cos^{2}\text{x}+4\sin^{2}\text{x}}$
→
Prove that the given vectors are non-coplanar:
$3\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}},\ 2\hat{\text{i}}-\hat{\text{j}}+7\hat{\text{k}}$ and $7\hat{\text{i}}-\hat{\text{j}}+23\hat{\text{k}}$
→
Solve the following systems of linear equations by cramer's rule:
x - 4y - z = 11,
2x - 5y + 2z = 39,
-3x + 2y + z = 1
→
Solve the following equation:
$\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=\text{y}^2$
→
If x = a (cos θ + log tan $\frac{\theta}{2}$) and y = a sin θ, find the value of $\frac{\text{dy}}{\text{dx}}$ at θ = $\frac{\pi}{4}.$
→