Answer to Question #237607 in Calculus for Anand

Question

Find the maximum height of the curve

Answer 1

A maximum or minimum value of a function is called it's extreme value. If

a function y=f(x) possess maximum and minimum, then

dy/dx=0 or f'(x)=0

The method used extensively to locate the Maxima and minima are as

Solve dy/dx=0 for x. suppose x=u is its one root.

(1) If second derivative of y i.e. d^2y/dx^2 is less than 0, then maximum value occur at x=u.

(2) If second derivative of y i.e. d^2y/dx^2 is greater than 0, then minimum value occur at x=u.

For example

y=-x²+ 6x-2

We find out the maximum height of the above quadratic equation curve at any point.

dy/dx=-2x+6

First we calculate value of x, by equating dy/dx=0

-2x+6=0

2x=6

x=6/2=3

i.e. dy/dx=0, at x=3

Second derivative d²y/dx²=-2

i.e it is less than 0 than maximum value of y occur at x=3

y=-x²+6x-2

Maximum height of the Quadratic equation curve occur at x=3

So y_max = -3²+6(3)-2=-9+18-2

y_max = -9+16=7