Match the LIST-I with LIST-II
| LIST-I Function | LIST-II Critical point | ||
| A. | \( y=x^2+x+1 \) | I. | 2/3 |
| B. | \( y=2x^3-6x^2+6x \) | II. | −1 |
| C. | \( y=2x^2+4x-1 \) | III. | −1/2 |
| D. | \( y=x^3-x^2 \) | IV. | 1 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → A-III, B-IV, C-II, D-I
A critical point is where \( \frac{dy}{dx}=0 \).
(A) \( y=x^2+x+1 \)
This matches III. −1/2.
\( \displaystyle \frac{dy}{dx}=2x+1=0\;\Rightarrow\;x=-\frac{1}{2} \)
(B) \( y=2x^3-6x^2+6x \)
This matches IV. 1.
\( \displaystyle \frac{dy}{dx}=6x^2-12x+6=6(x-1)^2 \)
\( \displaystyle 6(x-1)^2=0\;\Rightarrow\;x=1 \)
(C) \( y=2x^2+4x-1 \)
This matches II. −1.
\( \displaystyle \frac{dy}{dx}=4x+4=0\;\Rightarrow\;x=-1 \)
(D) \( y=x^3-x^2 \)
This matches I. 2/3.
\( \displaystyle \frac{dy}{dx}=3x^2-2x=x(3x-2) \)
\( \displaystyle x(3x-2)=0\;\Rightarrow\;x=0\ \text{or}\ x=\frac{2}{3} \)
Of these, 2/3 is the value listed.
So the matching is A-III, B-IV, C-II, D-I.