Give the signs (+ or - or 0)

Question

Give the signs (+ or - or 0) of 

y = f(x) (shown in the graph) and 

f '(x) and f''(x) when x = 4

The signs of f(4), f '(4) and f ''(4) are

Answer 1

It's easy to see that f(4) is zero (that's just the y value of the function when x = 4).

f'(4) is the slope of the tangent line to the curve at x = 4. I sketched it below:

The slope of that blue line is positive (+)

f''(4) is a measure of concavity. If f(x) is concave up, f'' is positive; if it's concave down, f'' is negative. Here, f(x) is obviously concave down so f''(4) is negative (-)