Question

2. Find the values of x for which the given curve
y = ax3 + bx2 + cx + d is concave upward and concave
downward.​

Answer 1

Answer:

y=x

4

+ax

3

+bx

2

+cx+d ( at least 3 points of intersection with x-axis)

Case 1: One point of intersection

(x−α)(kx

3

+βx

2

+αx+δ)=0

Then (kx

3

+βx

2

+αx+δ) won't have solution.

But cubic polynomial will yield one more real solution

Case 2: Three points of intersection

(x−α)(x−β)(x−δ)(mx+h)=0

Now, if there are 3 points of intersection

There must be one point which is either local maximum or minimum. As if there are 3 solutions, there will be a compulsory fourth solution.