The graph of #y=ax^2+bx# has an extremum at #(1,-2)#. Find the values of a and b?

Answer 1

#f(x)=2x^2-4x#

#f(x)=ax^2+bx# , #x##in##RR#
According to Fermat's Theorem #f'(1)=0#
but #f'(x)=2ax+b#
#f'(1)=0# #<=># #2a+b=0# #<=># #b=-2a#
#f(1)=-2# #<=># #a+b=-2# #<=># #a=-2-b#
So #b=-2(-2-b)# #<=># #b=4+2b# #<=>#
#b=-4#
and #a=-2+4=2#
so #f(x)=2x^2-4x#
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Answer 2

#a = 2# and #b=-4#

Given: #y=ax^2+bx, y(1) = -2#

From the given can substitute 1 for x and 2 for y and write the following equation:

#-2 = a+b" [1]"#
We can write the second equation using that the first derivative is 0 when #x =1#
#dy/dx= 2ax+b#
#0= 2a+b" [2]"#

Subtract equation [1] from equation [2]:

#0 - -2=2a+b - (a+b)#
#2 = a#
#a=2#
Find the value of b by substituting #a = 2# into equation [1]:
#-2 = 2+b#
#-4 = b#
#b = -4#
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Answer 3

To find the values of ( a ) and ( b ), we use the fact that the extremum of the parabola occurs at the vertex, which is given as (1, -2).

Given the general form of a quadratic function ( y = ax^2 + bx ), the x-coordinate of the vertex is ( -\frac{b}{2a} ).

From the given information, we know that the x-coordinate of the vertex is 1. So,

[ -\frac{b}{2a} = 1 ]

This implies ( b = -2a ).

Also, at the vertex, the y-coordinate is -2. So,

[ a(1)^2 + b(1) = -2 ] [ a + b = -2 ]

Substituting ( b = -2a ) into this equation:

[ a + (-2a) = -2 ] [ a - 2a = -2 ] [ -a = -2 ] [ a = 2 ]

Substitute ( a = 2 ) into ( b = -2a ):

[ b = -2(2) ] [ b = -4 ]

Therefore, the values of ( a ) and ( b ) are ( a = 2 ) and ( b = -4 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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