Question

If \(y=\alpha \log x+\beta x^3-x\) has extreme values at \(x=-1\) and \(x=1\), then \(\alpha\) and \(\beta\) are respectively

• A. 0 and \(\frac{1}{3}\)
• B. 0 and \(\frac{-1}{3}\)
• C. \(\frac{-1}{3}\) and \(\frac{1}{3}\)
• D. \(\frac{1}{3}\) and \(\frac{1}{3}\)

Hint:

Set the first derivative equal to zero at both points and solve the two linear equations.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
At an extreme value the derivative is zero.

Step 2: Differentiate
\(\dfrac{dy}{dx}=\dfrac{\alpha}{x}+3\beta x^{2}-1\).

Step 3: Apply the conditions
At \(x=1\): \(\alpha+3\beta-1=0\).
At \(x=-1\): \(-\alpha+3\beta-1=0\).

Step 4: Solve
Adding gives \(6\beta=2\Rightarrow\beta=\dfrac{1}{3}\); subtracting gives \(\alpha=0\).

Final Answer:
alpha is 0 and beta is 1/3. \[ \boxed{\text{(A) }0\text{ and }\dfrac{1}{3}} \]