Minimum value of \(\displaystyle\sqrt{{{x}^{{{4}}}+{3}{x}^{{{2}}}-{6}{x}+{10}}}+\sqrt{{{x}^{{{4}}}-{5}{x}^{{{2}}}+{9}}}\) without using calculus?

Miguel Hanson

Miguel Hanson

Answered question

2022-04-16

Minimum value of
x4+3x26x+10+x45x2+9 without using calculus?

Answer & Explanation

izvozna39g0

izvozna39g0

Beginner2022-04-17Added 9 answers

Step 1
Let y=x2, notice
x2+(y3)2=x2+(x23)2=x45x2+9
(x3)2+(y+1)2=(x3)2+(x2+1)2=x43x26x+10
The problem at hand can be rephrased as:
Givin A=(0, 3), B=(3, 1) and P=(x, y) be a point on the parabola y=x2. What is the minimum value of AP+PB?
If one make a plot of the parabola y=x2, one will notice the parabola intersect with the line segment AB, this means the minimum value of AP+PB is
AB=(03)2+(3(1))2=32+42=5

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