Flightpath of Hang Glider

Let's get $f'(x) = 4x^3-28x+4ax$ and set it to zero to get the local maxima

$x^3-7x+a=0 =>$

Using the formula: $x = \frac{7±\sqrt{49-4a}}{2}$

x doesn't exist if $49-4a < 0$ , thus for $a = [1, 12]$ x exists, the remaining values for $a$ are 999 - 12 = $\boxed{987}$ .

for the given equation to not have any local maximum, its derivative should not have a root where its changes its sign from positive to negative when going from left to right on the number line. First calculate f'(x), here f'(x)=4x^3 - 28x^2 + 4ax there fore the above equation should have only one real root or-else the condition mentioned above will not be satisfied.

there fore f'(x)=4x(x^2 - 7x +a)=0 should have only one root => x^2 - 7x +a has no roots => discriminant<0 => 7^2 - 4a < 0 => a > 49/4 => a > 12.25

there fore the number of integral values of a is 1000-13=987

In order to have no local maxima you have to make sure that the derivative cannot equal zero. The derivative is 4x^3 -28x^2 + 4ax. You can factor this to 4x(x^2 - 7x + a). x = 0 will make the derivative equal zero but that is at the beginning of the path. The trinomial does not factor so I looked at the discriminant which is 49 - 4a. If the discriminant is zero or positive then the trinomial will have values that make it equal zero. So making 49 - 4a < 0 will guarantee that no values of x will make the derivative equal zero. Solving you get a > 12.5. Meaning that if a is a positive integer it needs to be at least 13 and since a <1000 you have 987 possibilities.

987