Class 11Mathematics4-2.Quadratic Equations and InequationsSolution of quadratic equations and Nature of roots
MediumFind the maximum and minimum values of the function $f(x) = 9x^{2} + 12x + 2$.
(D) The given function is $f(x) = 9x^{2} + 12x + 2$.
We can rewrite this by completing the square:
$f(x) = (3x)^{2} + 2(3x)(2) + 2^{2} - 2^{2} + 2$
$f(x) = (3x + 2)^{2} - 4 + 2$
$f(x) = (3x + 2)^{2} - 2$.
Since $(3x + 2)^{2} \geq 0$ for all $x \in \mathbb{R}$,the minimum value of $(3x + 2)^{2}$ is $0$.
Therefore,the minimum value of $f(x)$ is $0 - 2 = -2$,which occurs when $3x + 2 = 0$,i.e.,$x = -\frac{2}{3}$.
As $x \to \infty$,$f(x) \to \infty$,so the function does not have a maximum value.
We can rewrite this by completing the square:
$f(x) = (3x)^{2} + 2(3x)(2) + 2^{2} - 2^{2} + 2$
$f(x) = (3x + 2)^{2} - 4 + 2$
$f(x) = (3x + 2)^{2} - 2$.
Since $(3x + 2)^{2} \geq 0$ for all $x \in \mathbb{R}$,the minimum value of $(3x + 2)^{2}$ is $0$.
Therefore,the minimum value of $f(x)$ is $0 - 2 = -2$,which occurs when $3x + 2 = 0$,i.e.,$x = -\frac{2}{3}$.
As $x \to \infty$,$f(x) \to \infty$,so the function does not have a maximum value.
Explore More
Similar Questions
Let $\alpha$ and $\beta$ be the roots of the quadratic equation $a x^2+b x+c=0$. Observe the lists given below:
List-$I$ List-$II$ $(i)$ $\alpha = \beta$ $(A)$ $(ac^2)^{1/3} + (a^2c)^{1/3} + b = 0$ $(ii)$ $\alpha = 2\beta$ $(B)$ $2b^2 = 9ac$ $(iii)$ $\alpha = 3\beta$ $(C)$ $b^2 = 6ac$ $(iv)$ $\alpha = \beta^2$ $(D)$ $3b^2 = 16ac$ $(E)$ $b^2 = 4ac$ $(F)$ $(ac^2)^{1/3} + (a^2c)^{1/3} = b$
The correct match of List-$I$ from List-$II$ is:
| List-$I$ | List-$II$ |
| $(i)$ $\alpha = \beta$ | $(A)$ $(ac^2)^{1/3} + (a^2c)^{1/3} + b = 0$ |
| $(ii)$ $\alpha = 2\beta$ | $(B)$ $2b^2 = 9ac$ |
| $(iii)$ $\alpha = 3\beta$ | $(C)$ $b^2 = 6ac$ |
| $(iv)$ $\alpha = \beta^2$ | $(D)$ $3b^2 = 16ac$ |
| $(E)$ $b^2 = 4ac$ | |
| $(F)$ $(ac^2)^{1/3} + (a^2c)^{1/3} = b$ |
The correct match of List-$I$ from List-$II$ is:
If $2+\sqrt{3}$ is a root of the equation $f(x)=x^4+2x^3-16x^2-22x+7=0$,then which one of the following is not a root of $f(x)=0$?
The number of distinct quadratic equations $ax^2 + bx + c = 0$ with unequal real roots that can be formed by choosing the coefficients $a, b, c$ such that $a \neq b \neq c$ from the set $\{0, 1, 2, 4\}$ is:
If $x = 2 + \sqrt{3}$,then what is the value of $x^3 - 7x^2 + 13x - 12$?
Medium
View SolutionIf $\alpha \neq 0$ and $0$ are the roots of the equation $x^2 - 5kx + (6k^2 - 2k) = 0$,then $\alpha = $
Vedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo