Question

Need help urgently :(

The graph of the derivative, f '(x) is shown below. On what interval is the graph of f (x) concave up?

graph is a parabola with x intercepts at x equals negative 4 and 2 and y intercept at y equals negative 2

Concave up on (−∞, −1)
Concave up on (−∞, −∞)
Concave up on (−1, ∞)
Concave up on (−6, −2)

Answer 1

(-1,inf)
Concave up, from a second derivative graph, is points above y=0, as the f'(x) graph is reflective of the tangent line's slope with the zeros of f'(x) being inflection points. f"(x) graph would be positive if concave up and negative if concave down, which is a lot easier to see than here. This graph would be increasing from (-1, inf), with decreasing from (-inf,-1). As this shows a negative slope before (zero) inflection point at x=-4, and continues to be negative before reaching x=-1, this portion would be concave down. After x=-1, the slope becomes positive, even after reaching inflection point, so this would be concave up. Therefore, (-1,inf) is the best choice 

Answer 2

check the picture below, ditto.