calculus question

Question

The graph of f '(x) is continuous and increasing with an x-intercept at x = 0. Which of the following statements is false? (4 points)

	The graph of f is always concave up.
	The graph of f has an inflection point at x = 0.
	The graph of f has a relative minimum at x = 0.
	The graph of the second derivative is always positive.

Answer 1

When x < 0, the derivative is negative.

So, the function is decreasing when x < 0.

When x > 0, the derivative is positive.

So, the function is increasing when x > 0.

When x = 0, the derivative is zero.

So, the slope of the tangent to the function at x = 0 is zero.

So, the function has a max or min at x = 0.

But, since the function is decreasing before x = 0 and increasing after,

There must be a minimum at x = 0.

So, the function is like a parabola with vertex at x = 0

And, the function opens up.

So, it's always concave up.

So, the second derivative is always positive.

All of the above statements are True except there is NOT an inflection at x = 0.

Inflections occur when the 2nd derivative = 0.

Since the 1st derivative is always increasing,

The 2nd derivative will always be positive and NOT zero.