Let a quadratic function fx-x2-bx-c have two distinct roots - - and - - Then the maximum value of gx-2fx-18fx in - is

Given : nbsp;f(x)=x^2+bx+c have two distinct roots Sign scheme for f(x) : ⇒ f(x) lt;0 nbsp;in (α,β) Putting, h(x)= f(x) gt;0 in (α,β) Now, g(x)=2f(x)+1 ...

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Nature of Roots

Let a quadrat...

Question

Let a quadratic function $f (x) = x^{2} + b x + c$ have two distinct roots $α, β$ and $α < β .$ Then the maximum value of $g (x) = 2 f (x) + \frac{18}{f (x)}$ in $(α, β),$ is

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Solution

Given : $f (x) = x^{2} + b x + c$ have two distinct roots
Sign scheme for $f (x) :$

$\Rightarrow f (x) < 0$ in $(α, β)$
Putting, $h (x) = - f (x) > 0$ in $(α, β)$
Now,
$g (x) = 2 f (x) + \frac{18}{f (x)} g (x) = - [- 2 f (x) + \frac{18}{- f (x)}] g (x) = - [2 h (x) + \frac{18}{h (x)}]$

Using $A . M . \geq G . M . : h (x) > 0$
$\frac{1}{2} [2 h (x) + \frac{18}{h (x)}] \geq \sqrt{2 h (x) \cdot \frac{18}{h (x)}} \Rightarrow 2 h (x) + \frac{18}{h (x)} \geq 2 \times 6 \Rightarrow - [2 h (x) + \frac{18}{h (x)}] \leq - 12 \Rightarrow g (x) \leq - 12$
$∴$ maximum value of $g (x)$ is $- 12.$

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Let a quadratic function f(x)=x2+bx+c have two distinct roots α,β and α<β. Then the maximum value of g(x)=2f(x)+18f(x) in (α,β), is

Let a quadratic function $f (x) = x^{2} + b x + c$ have two distinct roots $α, β$ and $α < β .$ Then the maximum value of $g (x) = 2 f (x) + \frac{18}{f (x)}$ in $(α, β),$ is