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Bunuel
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Does the graph of the function f(x) = ax^2 + bx + 7, where a, b and c 04 Oct 2021, 01:30
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55% (01:18) correct 45% (01:25) wrong based on 193 sessions

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Does the graph of the function \(f(x) = ax^2 + bx + 7\), where a, b and c are constants, have a maximum point?


(1) ​\(\sqrt{a^2} ≠a\)

(2) \(a + b < 0\)


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A
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Does the graph of the function f(x) = ax^2 + bx + 7, where a, b and c 04 Oct 2021, 18:37
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Does the graph of the function \(f(x) = ax^2 + bx + 7\), where a, b and c are constants, have a maximum point?

To know whether the graph of that function has a maximum point, we need to know whether a < 0 or a > 0
a < 0 -> yes (maximum point only
a > 0 -> no (minimum point only)

(1) ​\(\sqrt{a^2} ≠a\)
<=>\( a ≠ 0 \)
and a < 0 --> sufficient

(2) \(a + b < 0\)
=> no clues about a
=> insufficient

IMO: A
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Does the graph of the function f(x) = ax^2 + bx + 7, where a, b and c 05 Oct 2021, 03:45
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can you please explain the concept of maximum and minimum here?

why will there be a maximum pt only if a<0
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Does the graph of the function f(x) = ax^2 + bx + 7, where a, b and c 05 Oct 2021, 07:32
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Hi nanya18 ,

The given function f(x) is a quadratic function, hence a parabola. The only thing that determines whether the quadratic function opens upwards or opens downwards is the coefficient \(a\). We will only have a maximum point (which is the y-coordinate of the vertex to be specific) if it opens downwards, hence \(a < 0\). Hope that helps!
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Does the graph of the function f(x) = ax^2 + bx + 7, where a, b and c 16 Aug 2023, 11:57
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Bunuel
Does the graph of the function \(f(x) = ax^2 + bx + 7\), where a, b and c are constants, have a maximum point?


(1) ​\(\sqrt{a^2} ≠a\)

(2) \(a + b < 0\)


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Bunuel - I do not see any official solution of expert response. Can you please help with a solution?
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Does the graph of the function f(x) = ax^2 + bx + 7, where a, b and c 16 Aug 2023, 13:19
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Engineer1

Bunuel - I do not see any official solution of expert response. Can you please help with a solution?
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Hello Engineer1

Not an expert. While an expert replies, sharing my views on this question.

A bit of theory

The function \(ax^2 + bx + c\) is a quadratic equation. The graph of the function represents a parabola on an XY plane. The so-formed graph, i.e. that parabola, can either be an upward parabola or a downward parabola as shown below -

Attachment:
parabola.jpg
parabola.jpg [ 22.32 KiB | Viewed 1863 times ]

The coefficient of \(x^2\), i.e. the value of '\(a\)', determines whether a parabola is upward or downward.

  • If \(a\) is positive, the graph so formed represents an upward parabola. Such a graph will not have any upper bounds, i.e. the graph will not have a finite maximum value but will have a lower bound, i.e. will have a point of minima (finite minimum value).
  • If \(a\) is negative, the graph so formed represents a downward parabola. Such a graph will not have any lower bounds i.e. the graph will not have a finite minimum value but will have an upper bound i.e. will have a maximum (finite) upper value.

Back to the question

Bunuel
Does the graph of the function \(f(x) = ax^2 + bx + 7\), where a, b, and c are constants, have a maximum point?

(1) ​\(\sqrt{a^2} ≠a\)

(2) \(a + b < 0\)

Show more

\(f(x) = ax^2 + bx + 7\)

Question: Does the graph of the function \(f(x) = ax^2 + bx + 7\), where a, b, and c are constants, have a maximum point?

Inference: The graph will have a maximum point if the parabola is a downward parabola. Hence, if the value of \(a\) is negative, we can conclude that the graph of the function \(f(x) = ax^2 + bx + 7\) has a maximum point

Target Question: Is '\(a\)' negative ?

Statement 1

(1) ​\(\sqrt{a^2} ≠a\)

\(\sqrt{a^2} = |a|\)

​\(|a| ≠a\)

This indicates that \(a\) is negative. Hence, we have a definite answer to our target question and statement 1 is sufficient to answer the question.

Eliminate B, C, and E.

Statement 2

(2) \(a + b < 0\)

The value of '\(a\)' can positive or it can be negative. Hence, we cannot have a fixed answer to our target question.

Example:
1) a = 10; b = -100

\(a + b < 0\), in this case as 'a' is positive, the parabola is an upward parabola. We will not have a maximum point.

2) a = -10; b = 1

\(a + b < 0\), in this case as 'a' is negative , the parabola is an downward parabola. We will have a maximum point.

As we have two contradicting answers, the statement alone is not sufficient. Eliminate D.

Option A

Hope it helps !
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