Suppose f(x) is such a quadrant expression that it is positive for all real x . If g(x)=f(x)+f^ (x)+f^ (x) , then for anyreal x . Then for any real x .

Correct answer: B. g(x) > 0 Let f(x)=a x^2+b x+c, a > 0, b^2 0) Now, g(x)=a x^2+b x+c+2 a x+b+2 a=a x^2+(b+2 a) x+2 a+b+c Now, (b+2 a)^2-4 a(2 a+b+c) aligned & =b^2+4 a b+4 a^2-8 a^2-4 a b-4 a c & =b^2-4 a c-8 a^2 0 aligned

NDA 2018

Exam: NDA
Year: 2018
Subject: Mathematics
Chapter: Quadratic Equation

NDA 2018 · Mathematics · Quadratic Equation

Suppose f(x) is such a quadrant expression that it is positive for...

Suppose

f(x)

is such a quadrant expression that it is positive for all real

x

. If

g(x)=f(x)+f^{\prime}(x)+f^{\prime}(x)

, then for anyreal

x

. Then for any real

x

Options

A $\mathrm{g}(\mathrm{x}) < 0$
B $\mathrm{g}(\mathrm{x}) > 0$
C $\mathrm{g}(\mathrm{x})=0$
D $g(x) \geq 0$

Correct answer

B. $\mathrm{g}(\mathrm{x}) > 0$

Step-by-step solution

Let

f(x)=a x^2+b x+c, a > 0, b^2 < 4 a c \quad(\because f(x) > 0)

Now,

g(x)=a x^2+b x+c+2 a x+b+2 a=a x^2+(b+2 a) x+2 a+b+c

Now,

(b+2 a)^2-4 a(2 \mathrm{a}+\mathrm{b}+\mathrm{c})

\begin{aligned} & =b^2+4 a b+4 a^2-8 a^2-4 a b-4 a c \\ & =b^2-4 a c-8 a^2 < 0 \\ & g(x) > 0 \end{aligned}

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