Calculas III HW How to find curve concave upward

Question

Q1a) Find dy/dx  and d²y/dx².

x=t²+3,    y=t²+5t

Q1b) For which values of t is the curve concave upward? (Enter your answer using interval notation.)

I was able to find dy/dx and d²y/dx²

However for Q1b) I am unsure what is the value of t is when the curve concave upward

dy/dx= 2t+5/2t

 d²y/dx²= -5/4t³

Answer 1

A line is considered to be concave up if its second derivative is positive, and concave down if its second derivative is negative. Therefore, we just need to solve for values of t such that d²y/dx² is positive and negative.

Since the whole expression has a negative in front of it, and since t³ can be both positive and negative, the second derivative would be negative when t is positive, and positive when t is negative. In interval notation, we would just say that the curve is concave upwards when t < 0.