First slide

Maxima and minima

Question

If $f (x) = x^{2} + 2 b x + 2 c^{2} a n d g (x) = - x^{2} - 2 c x + b^{2}$ are such that min f(x) > max g(x), then the relation between b and c is

Moderate

A

$no relation$
B

$0 < c < b / 2$
C

$|c| < |b| \sqrt{2}$
D

$|c| > |b| \sqrt{2}$

Solution

$\begin{array}{l} f (x) = x^{2} + 2 b x + 2 c^{2} = (x + b)^{2} + 2 c^{2} - b^{2} \\ \Rightarrow min f (x) = 2 c^{2} - b^{2} \\ Also \begin{aligned} g (x) & = - x^{2} - 2 c x + b^{2} \\ = b^{2} + c^{2} - (x + c)^{2} \\ So & max g (x) = b^{2} + c^{2} \\ Since & min f (x) > max g (x) \\ So & 2 c^{2} - b^{2} > b^{2} + c^{2} \\ \Rightarrow & c^{2} > 2 b^{2} \Rightarrow | c | > \sqrt{2} | b | \end{aligned} \end{array}$

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