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If x is real, then the maximum and minimum values of the expression \( \Large \frac{x^{2}
-3x+4}{x^{2}+3x+4} \) will be:

A) 2,1
B) \( \Large 5, \frac{1}{5} \)
C) \( \Large 7, \frac{1}{7} \)
D) none of these.

Correct Answer:



C) \( \Large 7, \frac{1}{7} \)

Description for Correct answer:
Let \( \Large y=\frac{x^{2}-3x+4}{x^{2}+3x+4} \)

=> \( \Large \left(y-1\right)x^{2}+3 \left(y+1\right)x+4 \left(y-1\right)=0 \)

For x is real, \( \Large D \ge 0 \)

=> \( \Large 9 \left(y+1\right)^{2}-16 \left(y-1\right)^{2}\ge 0 \)

=> \( \Large -7y^{2}+50y-7\ge 0 \)

=> \( \Large 7y^{2}-50y+7 \le 0 \)

=> \( \Large \left(y-7\right) \left(7y-1\right) \le 0 \)

=> \( \Large y \le 7\ and\ y \ge \frac{1}{7} => \frac{1}{7} \le y \le 7 \)

Hence maximum value 7 and minimum is \( \Large \frac{1}{7} \).

Part of solved Quadratic Equations questions and answers : >> Elementary Mathematics >> Quadratic Equations

Comments

Similar Questions

1). The number of real solutions of the equation \( \Large |x^{2}+4x+3|+2x+5=0 \) are:

A). 1

B). 2

C). 3

D). 4

2). If the roots of the given equation:
\( \Large \left(\cos p-1\right)x^{2}+ \left(\cos p\right)x+\sin p = 0 \) are real, then:

A). \( \Large P \epsilon \left(- \pi ,0\right) \)

B). \( \Large P \epsilon \left(- \frac{ \pi }{2}, \frac{ \pi }{2} \right) \)

C). \( \Large P \epsilon \left(0, \pi \right) \)

D). \( \Large P \epsilon \left(0, 2 \pi \right) \)

3). The solution of the quadratic equation \( \Large \left(3|x|-3\right)^{2}=|x|+7 \) which belongs to the domain of definition of function \( \Large \gamma = \sqrt{x \left(x-3\right) } \) are given by:

A). \( \Large \pm \frac{1}{9}, \pm 2 \)

B). \( \Large -\frac{1}{9}, 2 \)

C). \( \Large \frac{1}{9}, -2 \)

D). \( \Large -\frac{1}{9}, -2 \)

4). The number of solution of \( \Large \frac{log 5 + log \left(x^{2}+1\right) }{log \left(x-2\right) }=2 \)

A). 2

B). 3

C). 1

D). none of these

5). If the expression \( \Large \left(mx-1+\frac{1}{x}\right) \) is always nonnegative, then the minimum value of m must be:

A). \( \Large -\frac{1}{2} \)

B). 0

C). \( \Large \frac{1}{4} \)

D). \( \Large \frac{1}{2} \)

6). The value of x in the given equation \( \Large 4^{x}-3^{x-\frac{1}{2}}=3^{x+\frac{1}{2}}-2^{2x-1} \) is:

A). \( \Large \frac{4}{3} \)

B). \( \Large \frac{3}{2} \)

C). \( \Large \frac{2}{1} \)

D). \( \Large \frac{5}{3} \)

7). The harmonic mean of the roots of equation \( \Large \left(5+\sqrt{2}x^{2}-14+\sqrt{5}\right)x+8+2\sqrt{5}=0 \) is:

A). 2

B). 4

C). 6

D). 8

8). For what value of \( \Large \lambda \) the sum of the squares of the roots of \( \Large x^{2}+ \left(2+\lambda\right)n-\frac{1}{2} \left(1+\lambda\right)=0 \) is minimum?

A). \( \Large \frac{3}{2} \)

B). 1

C). \( \Large \frac{1}{2} \)

D). \( \Large \frac{11}{4} \)

9). Given that \( \Large \tan \alpha \) and \( \Large \tan \beta \) are the roots of \( \Large x^{2}-px+q=0 \), then the value of \( \Large \sin^{2} \left(\alpha + \beta \right) \) is equal to:

A). \( \Large \frac{p^{2}}{p^{2}+ \left(1-q\right)^{2} } \)

B). \( \Large \frac{p^{2}}{p^{2}+q^{2}} \)

C). \( \Large \frac{q^{2}}{p^{2} \left(1-q\right)^{2} } \)

D). \( \Large \frac{p^{2}}{ \left(p+q\right)^{2} } \)

10). The number of values of K for which the system of equations \( \Large \left(K+1\right)x+84=4K \) and \( \Large Kx+ \left(K+3\right)y=3 K-1 \) has infinitely many solution, is:

A). 0

B). 1

C). 2

D). infinite