JEE Quadratic 1

Algebra Level 4

Find the number of real values of parameter a a for which the largest value of the quadratic function f ( x ) = x 2 + a x + 2 f(x) = x^{2} + ax +2 in the interval [ − 2 , 4 ] [-2,4] is 6.


The answer is 0.

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Ninad Akolekar by Ninad Akolekar , 23, India

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2 solutions

Since the given function is a parabola that opens upward, the largest value of the function on the given interval will have to occur at one (or both) of the endpoints.

Suppose the maximum occurs at x = − 2. x = -2. Then f ( − 2 ) = 4 − 2 a + 2 = 6 ⟹ a = 0 f(-2) = 4 - 2a + 2 = 6 \Longrightarrow a = 0 , which would give us f ( x ) = x 2 + 2. f(x) = x^{2} + 2. But at x = 4 x = 4 this function would have a value of f ( 4 ) = 18 f(4) = 18 , contradicting the supposition that the maximum occurs at x = 2. x = 2.

Next, suppose the maximum occurs at x = 4. x = 4. Then f ( 4 ) = 16 + 4 a + 2 = 6 ⟹ a = − 3 f(4) = 16 + 4a + 2 = 6 \Longrightarrow a = -3 , which would give us f ( x ) = x 2 − 3 x + 2. f(x) = x^{2} - 3x + 2. But at x = − 2 x = -2 this function would have a value of f ( − 2 ) = 12 f(-2) = 12 , contradicting the supposition that the maximum occurs at x = 4. x = 4.

Finally, if a maximum were to occur at both endpoints, then

f ( − 2 ) = f ( 4 ) ⟹ 4 − 2 a + 2 = 16 + 4 a + 2 ⟹ a = − 2 f(-2) = f(4) \Longrightarrow 4 - 2a + 2 = 16 + 4a + 2 \Longrightarrow a = -2 ,

which would give us the function f ( x ) = x 2 − 2 x + 2. f(x) = x^{2} - 2x + 2. But in that case we would have f ( − 2 ) = f ( 4 ) = 10 f(-2) = f(4) = 10 and not the desired maximum of 6. 6.

Thus there are no values of a a that will yield the desired result.

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Nice Solution.

Ninad Akolekar - 6 years, 3 months ago

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Thanks; nice question. I just wrote you a note a moment ago regarding your other question, so if you get the chance to read it that would be great.

Brian Charlesworth - 6 years, 3 months ago

@Ninad Akolekar I just have a question regarding another question of yours, "JEE Quadratics #2". As presently written, there is actually no value of k k that will yield two distinct real roots. Did you mean to write x 2 + k x + k x^{2} + kx + k instead? If so, then I think that the answer would in fact be 5 5 , which is not one of the answer options. Could you please check and see if the question is as you intended? Thanks; I prefer to avoid formally disputing a question so I thought I would try to notify you this way instead. :)

Edit: Sorry, if it were x 2 + k x + k x^{2} + kx + k then one of the answer options will work, (I have answered it correctly just a moment ago). So I'm guessing now that this is the function you intended to write.

Brian Charlesworth - 6 years, 3 months ago

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Oops! I actually made mistake while typing the equation. Thanks Brian for pointing out the mistake. I have edited the equation accordingly.

And, thanks once again for not disputing the question.

Ninad Akolekar - 6 years, 3 months ago

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No problem. The question looks good now. :)

Brian Charlesworth - 6 years, 3 months ago

Whenever the question asks for number of real values of a variable, the answer is either infinity or 0. Real values cannot be counted.

Rishik Jain - 4 years, 10 months ago
Rishik Jain
Aug 17, 2016

Whenever the question asks for number of real values of a variable, the answer is either infinity or 0. Real values cannot be counted.

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