A student Sumit is running on a playground along the curve given by \(y = x^2 + 7\). Another student Amit standing at a point with co-ordinates (3,7) on the playground wants to hit Sumit by paper ball, when Sumit is nearest to him.
Based on the above information answer the following.
The position of Sumit when Amit will hit the paper ball is :
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (1, 8)
Amit throws when Sumit is nearest to him. Sumit is at \((x, x^2 + 7)\) and Amit at (3, 7):
\(D^2 = f(x) = (x - 3)^2 + x^4\)
\(f'(x) = 4x^3 + 2x - 6 = 2(x - 1)(2x^2 + 2x + 3)\)
Since \(2x^2 + 2x + 3 \gt 0\) always, f′(x) = 0 only at x = 1, and \(f''(x) = 12x^2 + 2 \gt 0\), so the distance is least at x = 1.
\(y = 1^2 + 7 = 8\)
So Sumit is at (1, 8).