First slide

Functions (XII)

Question

$If f (x) = x^{2} + x + \frac{3}{4} and g (x) = x^{2} + a x + 1 be two real$ $functions, then the range of a for which g (f (x)) = 0 has no$ $real solution is$

Moderate

A

$(- \infty, - 2)$
B

$(- 2, 2)$
C

$(- 2, \infty)$
D

$(2, \infty)$

Solution

$\begin{array}{l} f (x) = x^{2} + x + \frac{3}{4} = {(x + \frac{1}{2})}^{2} + \frac{1}{2} \geq \frac{1}{2} \\ g (f (x)) = (f (x)^{2} + a f (x) + 1 \\ for g (f (x)) = 0 \\ a = - (f (x) + \frac{1}{f (x)}) \leq - 2 \end{array}$

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