SATHEE: Theory of Equations 4 Question 7

Theory of Equations 4 Question 7

8. Let $f (x)$ be a quadratic expression which is positive for all real values of $x$ . If $g (x) = f (x) + f^{'} (x) + f^{''} (x)$ , then for any real $x$

$(1990, 2 M)$

(a) $g (x) < 0$

(b) $g (x) > 0$

(d) $g (x) \geq 0$

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Answer:

Correct Answer: 8. (b)

Solution:

Let $f (x) = a x^{2} + b x + c > 0, \forall x \in R$

$\begin{aligned} \Rightarrow & a & > 0 \\ and & b^{2} - 4 a c & < 0 \end{aligned}$

$\begin{array}{ll} ∴ & g (x) = f (x) + f^{'} (x) + f^{''} (x) \\ \Rightarrow & g (x) = a x^{2} + b x + c + 2 a x + b + 2 a \\ \Rightarrow & g (x) = a x^{2} + x (b + 2 a) + (c + b + 2 a) \end{array}$

whose discriminant

$\begin{aligned} = (b + 2 a)^{2} - 4 a (c + b + 2 a) \\ = b^{2} + 4 a^{2} + 4 a b - 4 a c - 4 a b - 8 a^{2} \end{aligned}$

$= b^{2} - 4 a^{2} - 4 a c = (b^{2} - 4 a c) - 4 a^{2} < 0 [from Eq. (i)]$

$∴ g (x) > 0 \forall x$ , as $a > 0$ and discriminant $< 0$ .

Thus, $g (x) > 0, \forall x \in R$ .

Theory of Equations 4 Question 7

8. Let $f (x)$ be a quadratic expression which is positive for all real values of $x$ . If $g (x) = f (x) + f^{'} (x) + f^{''} (x)$ , then for any real $x$

Analytical & Descriptive Questions

Answer:

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Theory of Equations 4 Question 7

8. Let f(x) be a quadratic expression which is positive for all real values of x. If g(x)=f(x)+f′(x)+f′′(x), then for any real x

Analytical & Descriptive Questions

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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8. Let $f (x)$ be a quadratic expression which is positive for all real values of $x$ . If $g (x) = f (x) + f^{'} (x) + f^{''} (x)$ , then for any real $x$