Question

Use the derivative function, f ' ( x ) f ′ ( x ) , to determine where the function f ( x ) = − 4 x 2 + 13 x − 7 f ( x ) = - 4 x 2 + 13 x - 7 is increasing.

Answer 1

Given:

The function is

`f(x)=-4x^2+13x-7`

To find:

The interval on which the function is increasing.

Solution:

We have,

`f(x)=-4x^2+13x-7`

Differentiate with respect to x.

`f'(x)=-4(2x)+13(1)-(0)`

`f'(x)=-8x+13`

Equate f'(x)=0.

`-8x+13=0`

`-8x=-13`

`x=(-13)/(-8)`

`x=1.625`

The point x=1.625 divides the number line in two parts
`(-\infty, 1.625)\text{ and }(1.625, \infty)`.

f'(x) is positive for
`(-\infty, 1.625)`. It means the function is increasing on this interval.

f'(x) is negative for
`(1.625,\infty)`. It means the function is decreasing on this interval.

At x=1.625, f'(x) is 0. It is turning point so it will included in both intervals.

Therefore, the function is increasing on
`(-\infty, 1.625]`.