Question

For x=4 the functiom f(x) =x2+bx+c has the minimum value of_9.find c

Answer 1

`c = 25`

Explanation:

Vertex of a quadratic function:

Suppose we have a quadratic function in the following format:

`f(x) = ax^(2) + bx + c`

It's vertex is the point
`(x_(v), y_(v))`

In which

`x_(v) = -(b)/(2a)`

`y_(v) = -(\Delta)/(4a)`

Where

`\Delta = b^2-4ac`

If a>0, the vertex is a minumum point, that is, the minimum value happens at
`x_(v)`, and it's value is
`y_(v)`.

In this question:

`a = 1`

Minimum value at
`x = 4` means that
`x_v = 4`. So

`x_(v) = -(b)/(2a)`

`4 = -(b)/(2)`

`b = -8`

So

`f(x) = x^2 - 8x + c`

Minimum value of 9:

`y_v = 9`

So

`y_(v) = -(\Delta)/(4a)`

`9 = -(\Delta)/(4)`

`-\Delta = 36`

`\Delta = -36`

`b^2 - 4ac = -36`

`(-8)^2 - 4c = -36`

`-4c = -100`

`4c = 100`

`c = (100)/(4)`

`c = 25`