For the quadratic expression y=ax2+bx+c,a<0. The maximum value of y occurs at
A
x=−D4a2
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B
x=−b2a
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C
x=D4a2
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D
x=b2a
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Solution
The correct option is Bx=−b2a Given : y=ax2+bx+c;a<0
y=a(x2+bax+ca)
y=a(x2+2.b2ax+b24a2−b24a2+ca)
y=a((x+b2a)2−b24a2+4ac4a2)
y=a((x+b2a)2−(b24a2−4ac4a2))
y=a((x+b2a)2−(b2−4ac4a2))
y=a((x+b2a)2−(D4a2))
As a<0 and (x+b2a)2≥0
Therefore the maximum value occurs when the expression (x+b2a) becomes zero, i.e., x+b2a=0 ⇒x=−b2a
Conversly, we can take the graphical approach as well.
For that, let's draw the graph of any quadratic expression with a<0 as shown:
As clear from the graph, we can conclude that the expression attains it's maximum value at it's vertex or at x=−b2a.