SOLUTION: Given that y = √4x-7 - 2/3(x-4), find dy/dx and show that d^y/dx^2 = -4/(4x-7)√4x-7 . Hence find the maximum point of the curve.

Algebra ->  Test -> SOLUTION: Given that y = √4x-7 - 2/3(x-4), find dy/dx and show that d^y/dx^2 = -4/(4x-7)√4x-7 . Hence find the maximum point of the curve.      Log On


   

Question 1192427: Given that y = √4x-7 - 2/3(x-4), find dy/dx and show that d^y/dx^2 = -4/(4x-7)√4x-7 . Hence find the maximum point of the curve.
Answer by greenestamps(13195) About Me  (Show Source):
You can put this solution on YOUR website!


Note it is not good form to use the "√" symbol when writing your function -- we have to guess how much of the following expression is under the radical. Use "sqrt" with parentheses around the radicand.

y=sqrt%284x-7%29-%282%2F3%29%28x-4%29=%284x-7%29%5E%281%2F2%29-%282%2F3%29%28x-4%29

dy%2Fdx=%281%2F2%29%284%29%284x-7%29%5E%28-1%2F2%29-2%2F3=2%2Fsqrt%284x-7%29-2%2F3

The maximum or minimum point(s) are where the derivative is zero.

2%2Fsqrt%284x-7%29-2%2F3=0
2%2Fsqrt%284x-7%29=2%2F3
sqrt%284x-7%29=3
4x-7=9
4x=16
x=4

There is a single maximum or minimum, at x=4.

dy%2Fdx=2%284x-7%29%5E%28-1%2F2%29-2%2F3


The second derivative is always negative, so the graph has a local maximum at x=4.

f%284%29=sqrt%284x-7%29-%282%2F3%29%28x-4%29=sqrt%2816-7%29-%282%2F3%29%280%29=sqrt%289%29=3

ANSWER: The maximum point on the curve is (4,3)

A graph, showing (not very well) the maximum point at (4,3)

graph%28400%2C400%2C-1%2C7%2C-1%2C5%2Csqrt%284x-7%29-%282%2F3%29%28x-4%29%29