Question

3 The The curve y=ax² +bac + 5 where a and b are constants has a turning point p (1,3). find the values of a and b and determine whether P is a maximum or a minimum point,​

Answer 1

Answer:

a = 2, b = -2

P is at a minimum

Step-by-step explanation:

For any polynomial function, the turning point is either a maximum or a minimum

It can be determined by taking the first derivative of the function and setting it equal to 0 and solving for x and y

In this case we are given the turning point as x=1, y = 3 and we have to calculate a and b in the equation y = ax² + (ab)x + 5

The first derivative of y, y' = 2ax + ab

If we set this equal to 0 we get 2ax + ab = 0 ==> 2ax = -ab ==> b = -2

So the equation is of the form y = ax² -2ax + 5  (subbing for b)

Since we know that at x = 1, y =3 substitute y and x values in the above equation and solve for a

y = 3 = a(1²) - 2a(1) + 5

a - 2a + 5 = 3

-a + 5 = 3

a = 2

So the equation is of the form y = 2x² -4x + 5

If we plot this we will find that P is a minimum point

However, we can always determine mathematically if P is a max or min by taking the second derivative of the original function and noting the sign.  If it is positive, the point is a minimum, and if it is negative, the point is a maximum.

Taking derivatives,

y' = 4x  - 4

y'' = (4x-4)' = 4

The sign is positive so P is a minimum

Graph attached for reference