The maximum value of 12 x2-3x-2 isx -

Let f(x)=x^2 3x+2, a=1( gt;0), b= 3, c=2 Minimum value of f(x), 4ac b^24a= 14 ∴ Range of f(x) is [ 14, ∞ nbsp; ) ⇒ Range of ( 12 )^f(x) is ( 0, ( 12 )^ 14 ...

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Standard XII

Mathematics

Range of a Quadratic Expression

The maximum v...

Question

The maximum value of ${(\frac{1}{2})}^{x^{2} - 3 x + 2}$ is
$(x \in R)$

0

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\frac{1}{4}

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\sqrt[4]{2}

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not defined

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Solution

The correct option is C $\sqrt[4]{2}$
Let $f (x) = x^{2} - 3 x + 2$ ,
$a = 1 (> 0)$ , $b = - 3$ , $c = 2$
Minimum value of $f (x)$ ,
$\frac{4 a c - b^{2}}{4 a} = - \frac{1}{4}$

$∴$ Range of $f (x)$ is $[- \frac{1}{4}, \infty)$
$\Rightarrow$ Range of ${(\frac{1}{2})}^{f (x)}$ is $⎛ ⎜ ⎜ ⎝ 0, {(\frac{1}{2})}^{\frac{- 1}{4}} ⎤ ⎥ ⎥ ⎦ = (0, \sqrt[4]{2}]$

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The maximum value of (12)x2−3x+2 is (x∈R)

The correct option is C 4√2Let f(x)=x2−3x+2, a=1(>0), b=−3, c=2 Minimum value of f(x), 4ac−b24a=−14 ∴ Range of f(x) is [−14,∞) ⇒ Range of (12)f(x) is ⎛⎜ ⎜⎝0,(12)−14⎤⎥ ⎥⎦=(0,4√2]

The maximum value of ${(\frac{1}{2})}^{x^{2} - 3 x + 2}$ is
$(x \in R)$