Question

If $y = \alpha \log x + \beta x^2 - x$ has extreme values at $x = -1$ and $x = 1$, then $\alpha$ and $\beta$ are respectively ______.

• A. 0 and $\frac{1}{2}$
• B. 0 and $-\frac{1}{2}$
• C. $-\frac{1}{2}$ and $\frac{1}{2}$
• D. $\frac{1}{2}$ and $\frac{1}{2}$

Hint:

Extreme values always mean $\frac{dy}{dx} = 0$. When a problem gives multiple conditions but options are limited, quickly test just ONE boundary condition against the options to find the unique match instantly!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The function has "extreme values" (local maxima or minima) at specific points. This means the first derivative $y'$ must equal exactly 0 at those given $x$-values.
(Note: A true $\log x$ function is undefined for negative numbers. We proceed assuming standard algebraic evaluation of the derivative function as commonly tested).

Step 2: Detailed Explanation:
The given function is:
$y = \alpha \log x + \beta x^2 - x$
Find the first derivative with respect to $x$:
$y' = \frac{\alpha}{x} + 2\beta x - 1$
Since the function has an extreme value at $x = 1$, the derivative must be 0 at $x = 1$:
$y'(1) = \frac{\alpha}{1} + 2\beta(1) - 1 = 0$
$\alpha + 2\beta - 1 = 0$
$\alpha + 2\beta = 1$ --- (Equation 1)
Strictly evaluating the options against Equation 1 ($\alpha + 2\beta = 1$):
- Option (a): $\alpha = 0, \beta = 1/2 \implies 0 + 2(1/2) = 1$. (Matches perfectly)
- Option (b): $\alpha = 0, \beta = -1/2 \implies 0 - 1 = -1 \ne 1$.
- Option (c): $\alpha = -1/2, \beta = 1/2 \implies -1/2 + 1 = 1/2 \ne 1$.
- Option (d): $\alpha = 1/2, \beta = 1/2 \implies 1/2 + 1 = 3/2 \ne 1$.
Only Option (a) mathematically satisfies the necessary condition for an extremum at $x = 1$.

Step 3: Final Answer:
The values are 0 and $\frac{1}{2}$, matching option (a).