Download App

Maxima and Minima — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSMaxima and Minima5 Marks
Question
Find the absolute maximum and the absolute minimum value of the following functions in the given intervals:
$f(x) = 3x^4 - 8x^3 + 12x^2 - 48x + 25$ in $[0,3]$
✓

Answer

Given, $f(x) = 3x^4 - 8x^3 + 12x^2 - 48 + 25$
$\Rightarrow f'(x) = 12x^3- 24x^2 + 24x - 28$
For a local maximum or a local minimum, We must have $f'(x) = 0$
$\Rightarrow 12x^3- 24x^2 + 24x - 48 = 0$
$\Rightarrow x^3 - 2x^2+ 2x - 4 = 0$
$\Rightarrow x^2(x - 2) + 2(x - 2) = 0$
$\Rightarrow (x - 2)(x^2 + 2) = 0$
$\Rightarrow x - 2 = 0$ or $(x^2 + 2) = 0$
$\Rightarrow x = 2$
No, real root exists for $(x^2 + 2) = 0$
Thus, the critical points of f are $0, 2$ and $3.$
Now, $f(0) = 3(0)^4 - 8(0)^3+ 12(0)^2 - 48(0) + 25 = 25$
$f(2) = 3(2)^4 - 8(2)^3 + 12(2)^2 - 48(2) + 25 = -39$
$f(3) = 3(3)^4 - 8(3)^3+ 12(3)^2 - 48(3) + 25 = 1$
Hence, the absolute maximum value when $x = 0$ is $25$ and the absolute minimum value when $x = 2$ is $−39.$

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 Maxima and Minima→Maxima and Minima — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the equation of the plane passing through the point (-1, 3, 2) and perpendicular to each of the planes $\text{x + 2y +3z = 5 and 3x + 3y + z} = 0$
→
Solve the following differential equation:
$\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=\text{x}\log\text{x}$
→
Solve the following differential equation:$\sqrt{\text{1 + x}^{2}+\text{y}^{2}+\text{x}^{2}\text{y}^{2}}+\text{xy}\frac{\text{dy}}{\text{dx}}=0.$
→
Find the area enclosed by the curve $y = -x^2$ and the straight lilne $x + y + 2 = 0.$
→
Show that the following system of linear equation is inconsistent:
4x − 5y − 2z = 2
5x − 4y + 2z = −2
2x + 2y + 8z = −1
→
Differentiatet $\tan^{-1}\Big(\frac{x}{\sqrt{1-x^2}}\Big)$ with respect to $\sin^{-1}(2x\sqrt{1-x^2}).$
→
If $A=\left[\begin{array}{cc}0 & -\tan \frac{\alpha}{2} \\ \tan \frac{\alpha}{2} & 0\end{array}\right]$ and I is the identity matrix of order 2 , show that
$I+A=(I-A)\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]$
→
Find one-parameter families of solution curves of the following differential equation: (or solve the following differential equation)$\frac{\text{dy}}{\text{dx}}-\text{y}=\cos2\text{x}$
→
Evaluate the following integrals:
$\int\limits^1_{-1}|\text{x}\cos\pi\text{x}|\text{dx}$
→
Evaluate the following definite integral as limit of sum:
$\int\limits^{\text{b}}_{\text{a}}\text{x dx}$
→