Question

Let P(a, b) be a point on the parabola y = 4x - x² and it is the point nearest to the point A(-1, 4). Then (a + b) equals to

Answer 1

4

Answer 2

Let P(a, b) be a point on the parabola $y = 4x - x^2$. The distance squared from A(-1, 4) to P(a, b) is $d^2 = (a+1)^2 + (b-4)^2$. Since $b = 4a - a^2$, we have $d^2(a) = (a+1)^2 + (4a - a^2 - 4)^2 = (a+1)^2 + (a-2)^4$. To minimize $d^2$, we find the derivative: $\frac{d(d^2)}{da} = 2(a+1) + 4(a-2)^3$. Setting the derivative to 0: $2(a+1) + 4(a-2)^3 = 0 \implies a+1 + 2(a-2)^3 = 0$. By inspection, $a=1$ is a solution. For $a=1$, $b = 4(1) - (1)^2 = 3$. The point is P(1, 3). Then $a+b = 1+3 = 4$.