Suppose f(3)=2,f'(3)=1/2 and f'(x)>0 and f''(x)<0 for all x . is it possible f'(2)=1/3

Answer 1

f'(x) > 0 means that f is increasing

f"(x) < 0 means that f is concave down

Draw a graph that is increasing and concave down. Draw the tangent lines to the curve when x = 3 and when x = 2

Since f'(3) = 1/2, the slope of the tangent line with point of tangency (3, f(3)) is 1/2

Because of the shape of the graph of y = f(x), the tangent line to the curve with point of tangency (2, f(2)) is steeper than the tangent line with point of tangency (3, f(3) and thus has slope > 1/2

So, it is not possible for f'(2) to equal 1/3.