Answer to this question cannot be D, it should be B (II only).

To solve this question you don't need to calculate a, b, and c. We should put the given three points on the XY-plane and we'll get: parabola \(f(x)=ax^2+bx+c\) will have the vertex at \(x=1\), halfway between \(x=-3\) and \(x=5\), as \(f(-3)=0=f(5)\). Plus, as \(f(0)=3\) the parabola would be downward. As the parabola is downward, value of f(x) at x=1, f(1), is the the greatest value of f(x). As we move from x=1 to either of direction the value of f(x) will decrease. So any \(f(m)\) would be greater than \(f(n)\) if \(m\) is closer to \(x=1\) than \(n\). For example \(f(9)>f(10)>f(-12)\) or \(f(6)=f(-4)\) or \(f(-5)=f(7)>f(-100)\).

Hence
I. \(f(-1)>f(2)\) is false, as \(x=2\) is 1 farther from \(x=1\) and \(x=-1\) is 2 farther than \(x=1\). (True inequality would be \(f(-1)<f(2)\)).

II. \(f(1)>f(0)\) is true as \(f(1)\) is vertex and more than any \(f(x)\).

III. \(f(2)>f(1)\) is false as no \(f(x)\) is more than \(f(1)\). (The correct would be \(f(2)<f(1)\)).

Hope it's clear.
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f(x)=y

Substituting (x,y) as (-3,0) , (0,3) and (5,0), we get the following equations:

0 = 9a-3b+c
3 = c
0 = 25a+5b+c

Solving: a=-1/5, b=2/5, c=3
f(x) = -x^2/5 + x/5 +3

The options are on f(-1), f(0), f(1) and f(2).
f(-1) = -1/5 + 2/5 + 3 = 16/5 = 3.2
f(0) = 3
f(1) = 1/5 + 2/5 + 3 = 18/5 = 3.6
f(2) = -4/5 + 4/5 + 3 = 3

I f(-1) > f(2) true
II f(1) > f(0) true
III f(2) > f(1) false

So, option D, I and II

(not sure if there is an easier way to solve this!)

@bunnuel i have dited my post for the st I ..it was a typo error i am sorry abt that

it is f(-1) > f(-2)

i am not clear as to how did u get the vertex as x=1

Intersection points of parabola with x-axis are \((-3,0)\) and \((5,0)\):

-----(-3)--0----(5)---, as parabola is symmetric, the x coordinate of the vertex must be halfway between \(x=-3\) and \(x=5\) --> \(x=\frac{-3+5}{2}=1\). As parabola is downward, \(f(x)\) naturally will have it's max values at vertex \(x=1\), \(f(1)\).

As for the typo in the stem. If I is saying: \(f(-1) > f(-2)\), then it's true as \(x=-1\) is closer to \(x=1\), than \(x=-2\), which means that \(f(-1)>f(-2)\).
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amazing explanation Bunuel. I would normally have found a, b and c and then solved. Thanks for showing me the way to think more than the way you solved the problem.

HI Bunuel,

Please explain the red part , what i understood from making graph , if we move left from the axis point certainly value of f(x) decreases but if we move right side from the axis where x =1, value of f(x) will increase till certain point and then will decrease, so how can we deduce that value of f(x) is max at x =1

Check below:


Also check parabola chapter HERE.

Hope it helps.
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Wonderful approach! thanks a lot!

Hi Bunuel, could you please explain the highlighted part. How do we infer that parabola is downward? According to theory we only know that parabola is downward if a<0.
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Hint: put the given three points on the plane,
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I got it. We can know it when putting all the 3 points on a plane. I thought there's another special way. Thanks
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Concept: Once you find the Line of Symmetry of a Downward Opening Parabola, the X Coordinate at this Vertex will give you the HIGHEST VALUE that f(x) = y can have


Rule: to find the Line of Symmetry of a Parabola (in which everything on the Left Side of the Axis/Line is a "MIRROR REFLECTION" of all the Points on the Right Side), find the MID-POINT between the X-Intercepts that cross the X-Axis


between (-3 , 0) and (+5 , 0) ---- the Line of Symmetry will occur at the Vertical Line of: X = +1

Also, since this is a Downward Opening Parabola, the Vertex at f(+1) will also provide the HIGHEST VALUE of y

I. f(-1) > f(2)


f(-1) is +2 Units to the LEFT of the Highest Value at the Vertex f(+1)

However, f(2) is only +1 Unit to the RIGHT of the Highest Value at the Vertex f(+1)

the Value of f(2) will be GREATER than the Value of f(-1)

NOT True



II. f(1) > f(0)

The Highest Value will occur at the Vertex, when X =+1, because this is a Downward Opening Parabola.

Thus f(1) will be greater than any other Value

TRUE


III. f(2) > f(1)

As stated above, the Highest Value will occur at the Vertex when the X-Coordinate is +1 (at the Line of Symmetry) because this is a Downward Opening Parabola.

thus: f(2) < f(1)

III is NOT True


ONLY II must be True

-B-