Mathematics: What is the maximum vertical distance between the line y=x+2 and parabola y=x^2 for -1 =x =2

If D 0, the roots of the quadratic equation will b complex.Step-by-step explanation:Let is the given quadratic equation. So, if the Discriminant (D) = , the the equation will have only complex solutions.Complex roots are represented in the form of 1. x represents the X axis (real) co ordinate, y represents the Y axis (imaginary) coordinate2. Also, the graph of the with equation D 0 NEVER CROSSES the X - axis, as there are no real roots of the equation, only complex roots. 3. Here. roots...

What is the maximum vertical distance between the

Ответ:

olgapagan3173

01.01.2022

Mathematics

$x^2=x+2\implies x^2-x-2=(x-2)(x+1)=0\implies x=-1,x=2$

are the intersection points of the line and parabola. x^2

is convex, which guarantees that $x+2\ge x^2$ over this interval. This means the vertical distance between the two functions is

d(x)=|(x+2)-x^2|=2+x-x^2

Differentiating, we have

d'(x)=1-2x

which has one critical point at

$1-2x=0\implies x=\dfrac12$

At this point, we have

$d\left(\dfrac12\right)=\dfrac94$

as the maximum vertical distance.