For what value of the constant c is the function f continuous on (−[infinity],[infinity]) ? f(x)={cx2+2xifx<3x3−cxifx≥3

c = 6.25

Step-by-step explanation:

We are given the following piecewise function:

Continuous function:

A function f(x) is continuous, at a point a, if:

exists and

In this question:

Piece-wise function, so we have to verify if the limit exists.

The function changes at x = 3. So we verify at a = 3.

It will exist if:

To the left:

Less than 3.

To the right:

Greater than 3.

f continuous:

They have to be equal:

alright so i think for this one it would to draw it out. this may be hard for me to try and draw, but i will do my best on here:

        r

  /       |     \

notice how it forms 2 triangles inside one bigger triangle.

so if we know the angle spr is the same as psr that is saying the measure of angle p is the same as angle s. this means the big triangle psr is either isosceles or equilateral. in both cases, the triangle is perfectly symmetrical when split down the middle, which is in this case rq. because we know it is perfectly symmetrical on both sides, we know that the two triangles inside the bigger one are equal.