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.
If D is the distance between Sumit and Amit, then the value of \(x\) for which D is minimum, is :
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 1
Sumit is at \((x, x^2 + 7)\) and Amit at (3, 7), so
\(D^2 = (x - 3)^2 + x^4\)
D is least exactly when \(D^2\) is least, so let \(f(x) = (x - 3)^2 + x^4\).
\(f'(x) = 2(x - 3) + 4x^3\)
\(= 4x^3 + 2x - 6\)
\(= 2(x - 1)(2x^2 + 2x + 3)\)
The factor \(2x^2 + 2x + 3\) is always positive (its discriminant 4 − 24 is negative), so f′(x) = 0 only at x = 1.
\(f''(x) = 12x^2 + 2 \gt 0\)
So, by the second derivative test, D is minimum at x = 1.