Now the second derivative of y= x² is 2 which is >0 so
the first derivative is always increasing.
That means y=x² increases at an increasing rate when dy/dx>0
Now for x <0 dy/dx = 2x is less than zero too, thus it is negative
at x=-3 the derivative y=2x is -6
at x=-2 the derivative y=2x is -4
and
at x=-1 the derivative y=2x is -2
and we see that the (first) derivative is decreasing in absolute value.
It is actually increasing, however on the negative side of the number
line, increasing means going from more negative to less negative, so
positive second derivative means an increasing first derivative.
When that first derivative is positive, that means the function is INCREASING at
an increasing rate
When that first derivative is negative, that means the function is DECREASING,
also literally at an increasing rate, but that means the rate of decrease become less steep.
In both cases, that means the function is concave UP and that any extrema identified by the
first derivative test would have to be minimum.
A negative second derivative implies just the opposite. If the second derivative is less than zero
then the first derivative is decreasing.
If it is positive, then it is increasing at a decreasing rate and therefore becoming less steep.
If the first derivative is negative, then the function is decreasing, but at a decreasing rate when
on the negative side, means the ABSOLUTE value of the negative derivative is increasing and
the function function is becoming more steep in the negative direction.
Both of these correspond to a function whose graph opens down or is concave down and
any extrema occurring at a point where the second derivative is negative are maximum values.
Q3) I know usually when after you get the 1st or 2nd derivative of the function, we have to find the x-values. What do those x-values mean?