If the minimum value of f(x)=2x^2+alpha x+8 i | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The minimum value of a quadratic function $f(x)=ax^2+bx+c$ (where $a>0$) is given by $-\frac{D}{4a} = \frac{4ac-b^2}{4a}$.
For $f(x)=2x^2+\alp

Vedclass · Accepted Answer

$\frac{160}{27}$

Vedclass · Answer

(B) The minimum value of a quadratic function $f(x)=ax^2+bx+c$ (where $a>0$) is given by $-\frac{D}{4a} = \frac{4ac-b^2}{4a}$.
For $f(x)=2x^2+\alpha x+8$, we have $a=2, b=\alpha, c=8$. The minimum value is $\frac{4(2)(8)-\alpha^2}{4(2)} = \frac{64-\alpha^2}{8}$.
The maximum value of a quadratic function $g(x)=ax^2+bx+c$ (where $a < 0$) is given by $-\frac{D}{4a} = \frac{4ac-b^2}{4a}$.
For $g(x)=-3x^2-4x+\alpha^2$, we have $a=-3, b=-4, c=\alpha^2$. The maximum value is $\frac{4(-3)(\alpha^2)-(-4)^2}{4(-3)} = \frac{-12\alpha^2-16}{-12} = \frac{12\alpha^2+16}{12}$.
Equating the two values:
$\frac{64-\alpha^2}{8} = \frac{12\alpha^2+16}{12}$
Multiplying both sides by $24$:
$3(64-\alpha^2) = 2(12\alpha^2+16)$
$192-3\alpha^2 = 24\alpha^2+32$
$160 = 27\alpha^2$
$\alpha^2 = \frac{160}{27}$.
Thus, option $(B)$ is correct.

If the minimum value of $f(x)=2x^2+\alpha x+8$ is the same as the maximum value of $g(x)=-3x^2-4x+\alpha^2$, then $\alpha^2=$

Similar Questions

If an angle $A$ of a $\Delta ABC$ satisfies $5 \cos A + 3 = 0$,then the roots of the quadratic equation $9x^2 + 27x + 20 = 0$ are

If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2 + x \sin \theta - 2 \sin \theta = 0$,where $\theta \in (0, \frac{\pi}{2})$,then the value of $\frac{\alpha^{12} + \beta^{12}}{(\alpha^{-12} + \beta^{-12})(\alpha - \beta)^{24}}$ is equal to

For $x \in R$,the number of real roots of the equation $3x^2 - 4|x^2 - 1| + x - 1 = 0$ is:

If $a, b, c$ are real and $x^3 - 3b^2x + 2c^3$ is divisible by $x - a$ and $x - b$,then:

The minimum value of $f(x) = \frac{x^2-2x+3}{x^2-4x+7}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module