f''\left(-\frac{1}{3}\right) > 0. Hence there is a local minimum when x=-\dfrac{1}{3}.

f''(1) =4. Hence f''(c) > 0 for critical value c and there is a local minimum when x=1.

f''\left(-\frac{1}{3}\right) =-3. Hence f''(c) < 0 for critical value c and there is a local maximum when x=-\dfrac{1}{3}.

f''(1) < 0. Hence there is a local maximum when x=1.

f''\left(-\frac{1}{3}\right) =-4. Hence f''(c) < 0 for critical value c and there is a local maximum when x=-\dfrac{1}{3}.

f''(1) =3. Hence f''(c) > 0 for critical value c and there is a local minimum when x=1.

When  x=1  there is a local maximum and the second derivative is positive.

 When  x=0  the second derivative of the function is positive.

When  x=-1  there is a local minimum and the second derivative is negative.

As  x  increases, there is a concavity change in the graph of the function from concave up to concave down.

When  x=-1  there is a local minimum and the second derivative is positive.

When  x=0  the first derivative will be negative.

 As  x  increases, there is a concavity change in the graph of the function from concave down to concave up.

When  x=1  there is a local maximum and the second derivative is negative.

negative at a critical point x=c, then the graph of the function is concave down and the function has a local minimum at x=c.

positive at a critical point x=c, then the graph of the function is concave up and the function has a local minimum at x=c.

positive at a critical point x=c, then the graph of the function is concave down and the function has a local maximum at x=c.

zero at a critical point x=c, then the graph of the function is neither concave down nor concave up at x=c.

negative at a critical point x=c, then the graph of the function is concave down and the function has a local maximum at x=c.

negative at a critical point x=c, then the graph of the function is concave up and the function has a local minimum at x=c.

f(x) is increasing.

f'(x) is increasing.

f''(x) is negative.

f(x) is positive.

f'(x) is positive.

f''(x) is positive.