Maths · Applications of derivatives: Rate of change of quantities, monotonic-Increasing and decreasing functions, Maxima and minima of functions of one variable
The point on the curve which is nearest to (3,0) is
The point on the curve \( y=x^{2} \) which is nearest to (3,0) is
- A. (1,-1)
- B. (-1,1)
- C. (-1,-1)
- D. (1,1)
Step-by-step solution
The distance squared from point (x, x^2) on the curve to (3,0) is d^2 = (x-3)^2 + x^4. Minimizing this, we differentiate: f'(x)=4x^3+2(x-3)=2(2x^3+x-3)=0. Solving gives x=1 (since 2+1-3=0). Second derivative positive confirms minimum. Thus the nearest point is (1,1).
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