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Standard XII

Mathematics

Property 4

For the expre...

Question

For the expression f(x) = a $x^{2}$ + bx + c (a > 0), $b^{2}$ > 4ac always. A real value $x_{0}$ will lie in between the roots of f(x) if

f(x₀) < 0

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f(x₀) > 0

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f(x₀) = 0

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f(x₀) ≥ 0

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Solution

The correct option is A
f(x₀) < 0

Here a > 0. So, the graph of f(x) will be in cup shape.

Visualizing the shape through graph,

Let $α$ , $β$ be the roots,

Observe that for any value $x_{0}$ on x-axis in between $α$ and $β$ , f( $x_{0}$ ) < 0

So, f( $x_{0}$ ) < 0 is required condition

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For the expression f(x) = a x2 + bx + c (a > 0), b2 > 4ac always. A real value x0 will lie in between the roots of f(x) if

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For the expression f(x) = a $x^{2}$ + bx + c (a > 0), $b^{2}$ > 4ac always. A real value $x_{0}$ will lie in between the roots of f(x) if

The correct option is A
f(x₀) < 0

Here a > 0. So, the graph of f(x) will be in cup shape.

Visualizing the shape through graph,

Let $α$ , $β$ be the roots,

Observe that for any value $x_{0}$ on x-axis in between $α$ and $β$ , f( $x_{0}$ ) < 0

So, f( $x_{0}$ ) < 0 is required condition