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MCQMediumJEE 2023Applications of Derivatives (Monotonicity, Extrema)

JEE Mathematics 2023 Question with Solution

Let x=2x = 2 be a local minima of the function f(x)=2x4−18x2+8x+12,x∈(−4,4).f(x) = 2x^4 - 18x^2 + 8x + 12, \quad x \in (-4, 4). If MM is the local maximum value of the function f(x)f(x) in (−4,4)(-4, 4), then MM is:

  • A

    126−33212\sqrt{6} - \frac{33}{2}

  • B

    126−31212\sqrt{6} - \frac{31}{2}

  • C

    186−33218\sqrt{6} - \frac{33}{2}

  • D

    186−31218\sqrt{6} - \frac{31}{2}

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: f(x)=2x4−18x2+8x+12f(x) = 2x^4 - 18x^2 + 8x + 12 on (−4,4)(-4,4).

Find: The local maximum value MM of f(x)f(x).

From the extracted solution,

f′(x)=8x3−36x+8=4(2x3−9x+2)f'(x) = 8x^3 - 36x + 8 = 4(2x^3 - 9x + 2)

For critical points,

f′(x)=0f'(x) = 0

The solution gives

x=26−2x = \frac{2}{\sqrt{6} - 2}

and then evaluates the local maximum at

x=26−2x = \frac{2}{\sqrt{6} - 2}

Further, the solution rewrites the function as

f(x)=(x2−2x−29)(2x2+4x−1)+24x+7.5f(x) = \left(x^2 - 2x - \frac{2}{9}\right)(2x^2 + 4x - 1) + 24x + 7.5

Using the value substituted in the source solution, it concludes

f(26−2)=M=126−332f\left(\frac{2}{\sqrt{6} - 2}\right) = M = 12\sqrt{6} - \frac{33}{2}

Therefore, the local maximum value is stated as 126−33212\sqrt{6} - \frac{33}{2}.

However, the same the solution explicitly marks The Correct Option is B, while option B is 126−31212\sqrt{6} - \frac{31}{2}. Following the instruction that the solution is the primary source for the answer label, the correct option is taken as B.

Noted source discrepancy

The solution's contains an internal inconsistency:

  1. It explicitly says The Correct Option is B.
  2. Its final computed expression is written as 126−33212\sqrt{6} - \frac{33}{2}, which matches option A.

Because the answer-resolution rule gives priority to the solution's stated conclusion label, the recorded answer is B. The numerical-expression mismatch should be treated as a source discrepancy.

Common mistakes

  • Students may differentiate 2x4−18x2+8x+122x^4 - 18x^2 + 8x + 12 incorrectly, especially the term −18x2-18x^2. The correct derivative is −36x-36x, not −18x-18x. Always apply the power rule carefully to every term.

  • A common error is to confuse a critical point with a local maximum automatically. Solving f′(x)=0f'(x)=0 only gives candidate points; one must still identify which critical point corresponds to the local maximum in the interval.

  • Students may trust the option expression without checking the source working, or trust the final expression without checking the stated option label. Here the source itself is inconsistent, so both the computed value and the declared option must be compared carefully.

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