If α, β are the roots of the

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Target Exam

CUET

Subject

-- Mathematics - Section A

Chapter

Applications of Derivatives

Question:

If α, β are the roots of the quadratic equation $x^2-(a-2)x-(a+1)=0$, where a is a variable then the least value of $α^2+β^2$ is:

Options:

None of these

Correct Answer:

Explanation:

$α+β=a-2,\,αβ=-(a+1)$ $S=α^2+β^2=(α+β)^2-2αβ=(a-2)^2+2(a+1)=a^2-2a+6$

$\frac{dS}{da}=2a-2$ For max/min. $\frac{dS}{da}=0⇒a=1$

Least value of $α^2+β^2$ is 5.