COMEDK · Maths · 27. Application of Derivatives

If the function \(f(x) = x^4 - 31x^2 + ax + 5\) has a turning point at \(x = 1\), then the value of '\(a\)' is _ and the function attains a _ at \(x = 1\)

A \(a = -50\) , local maxima
B \(a = 58\) , local maxima
C \(a = 50\) , local minima
D \(a = 58\) , local minima

Verified Solution

Answer & Solution

Correct Answer

(B) \(a = 58\) , local maxima

Step-by-step Solution

Detailed explanation

Given \(f(x) = x^4 - 31x^2 + ax + 5\) Differentiating with respect to \(x\), we get: \(f'(x) = 4x^3 - 62x + a\) Since \(f(x)\) has a turning point at \(x = 1\), \(f'(1) = 0\) \(4(1)^3 - 62(1) + a = 0\) \(a = 58\) Now, finding the second derivative: \(f''(x) = 12x^2 - 62\) At…

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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If the function \(f(x) = x^4 - 31x^2 + ax + 5\) has a turning point at \(x = 1\), then the value of '\(a\)' is ___ and the function attains a ___ at \(x = 1\)

Step-by-step Solution

If the function \(f(x) = x^4 - 31x^2 + ax + 5\) has a turning point at \(x = 1\), then the value of '\(a\)' is _ and the function attains a _ at \(x = 1\)