COMEDK · Maths · 27. Application of Derivatives

The curve \(4 y=3 x^4-2 x^2\) attains -------- at the points \(x=-\dfrac{1}{\sqrt{3}}\) and \(x=\dfrac{1}{\sqrt{3}}\)

A a minimum value and a maximum value respectively
B both minimum values
C a maximum value and a minimum value respectively
D both maximum values

Verified Solution

Answer & Solution

Correct Answer

(B) both minimum values

Step-by-step Solution

Detailed explanation

Given the curve \(4y = 3x^4 - 2x^2\), we have \(y = \dfrac{3}{4}x^4 - \dfrac{1}{2}x^2\). To find the critical points, we calculate the first derivative with respect to \(x\): \(\dfrac{dy}{dx} = \dfrac{3}{4}(4x^3) - \dfrac{1}{2}(2x) = 3x^3 - x = x(3x^2 - 1)\). Setting…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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