00:01
In this problem, we want to find the smallest integer a such that the intermediate value theorem guarantees that f of x has a 0 on the interval 0a, where f of x is defined as the parabola minus 8x squared plus 8x plus c.
00:22
So we want to find a 0 between 0 and a.
00:26
So let's first by finding our function zeros.
00:29
So our zeros correspond to locations where f of x is equal to 0.
00:45
This is the case when minus 8 x square plus 8x plus 6 is equal to 0.
00:57
Just divide everywhere by a factor 2, just to simplify, we have minus 4 x square plus 4 plus 3 equal to 0.
01:10
Now we can use our quadratic equation to solve for the 0.
01:13
We will have that our parabola will have zeros at x is equal to minus 4 plus or minus 4 square minus 4 times minus 4 times 3 all this divided by minus 4 times evaluating this expression we will have that a function has two zeros the first at minus one half and the second at three halves since we're only interested in the positive interval in zero and a we're only interested in this zero here so let's draw a graph to help us graphically what is going on let's draw a parabola with two zeros, one between minus one half and another zero at three halves.
02:51
And our parabola is going to be inverted because we have a negative number in front of our x squared term.
02:59
So our parabola is going to look something like this.
03:12
So between zero and three halves, f of x is positive...