For a quadratic expression y-fx-ax2-bx-c- if f- - 0 and for the rest of values of x - fx-0-Then a - and D -

F(x)=ax^2+bx+c, f(α) = 0 and f(x) lt;0 ∀ x∈ℝ α It is given that f(x) lt;0 ∀ x∈ℝ α ⇒ a lt;0, Also f(x)=0 only at x=α ⇒α is the only root f(x)=0 ⇒ D=0. ...

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Byju's Answer

Standard XII

Mathematics

Graphs of Quadratic Equation for different values of D when a<0

For a quadrat...

Question

For a quadratic expression $y = f (x) = a x^{2} + b x + c,$ if $f (α) = 0$ and for the rest of values of $x, f (x) < 0.$
Then $a$ , and $D$ .

= 0

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> 0

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\in R

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< 0

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Solution

The correct option is D $< 0$
$f (x) = a x^{2} + b x + c,$
$f (α) = 0$ and $f (x) < 0 \forall x \in R - α$

It is given that $f (x) < 0 \forall x \in R - α \Rightarrow a < 0,$

Also $f (x) = 0$ only at $x = α$ $\Rightarrow α$ is the only root $f (x) = 0$
$\Rightarrow D = 0$ .

The same conditions can be visualized using the graph as:

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For a quadratic expression y=f(x)=ax2+bx+c, if f(α)=0 and for the rest of values of x, f(x)<0. Then a , and D .

The correct option is D <0f(x)=ax2+bx+c, f(α)=0 and f(x)<0 ∀ x∈R−α It is given that f(x)<0 ∀ x∈R−α⇒a<0, Also f(x)=0 only at x=α ⇒α is the only root f(x)=0 ⇒D=0. The same conditions can be visualized using the graph as:

For a quadratic expression $y = f (x) = a x^{2} + b x + c,$ if $f (α) = 0$ and for the rest of values of $x, f (x) < 0.$
Then $a$ , and $D$ .