Question

Identify the interval(s) on which the function y = -x2 + 5x + 14 is positive.

Answer 1

Given:

$y=-x^2+5x+14$

To find:

The interval on which the given function is positive.

Solution:

We have,

$y=-x^2+5x+14$

Splitting the middle term, we get

$y=-x^2+7x-2x+14$

$y=-x(x-7)-2(x-7)$

$y=(x-7)(-x-2)$

$y=-(x-7)(x+2)$

Using zero product property, we get

$x-7=0\text{ and }x+2=0$

$x=7\text{ and }x=-2$

-2 and 7 divide the number line is three interval.

Interval             Signs of $y=-(x-7)(x+2)$         Result

(-∞,-2)                             (-)(-)(-)                           Negative    

(-2,7)                              (-)(-)(+)                           Positive    

(7,∞)                               (-)(+)(+)                          Negative    

So, the function is positive for the interval (-2,7), i.e., -2 < x < 7.

Therefore, the correct option is D.