Question

If \[ y=a\log|x|+bx^2+x \] has its extremum values at \[ x=-1 \quad\text{and}\quad x=2, \] then \((a,b)=\)

• A. \[ \left(2,\frac12\right) \]
• B. \[ \left(-2,\frac12\right) \]
• C. \[ \left(-2,-\frac12\right) \]
• D. \[ \left(2,-\frac12\right) \] \bigskip

Hint:

If a function has extrema at given points, immediately use \[ f'(x)=0 \] at those points. This usually gives a system of equations for the unknown parameters.

Answer 1

The Correct Option is D

Concept: Extrema imply derivative zero at those points. Differentiate, set up equations, solve for a and b.

Step 1: \(y = a\log|x| + bx^2 + x\). \(y' = a/x + 2bx + 1\). Extrema at \(x=-1,2\): \(y'(-1)=0, y'(2)=0\).

Step 2: At \(x=-1\): \(-a -2b + 1 = 0 \Rightarrow a+2b=1\). At \(x=2\): \(a/2 + 4b + 1 = 0 \Rightarrow a+8b=-2\).

Step 3: Subtract: \(6b = -3 \Rightarrow b=-1/2\). Then \(a + 2(-1/2)=1 \Rightarrow a=2\).

Step 4: Pair \((a,b) = (2,-1/2)\).

Step 5: Write the final answer. \(\boxed{\left(2,-\frac12\right)}\)