... = has solutions on [π , 6π ] f ′ (x) = has solutions on [π , 6π ] c f(x) = x sin x is a counterexample Consider the interval [ 0, π ] f ( −π ) = f (π ) = and f ( x ) = has exactly two solutions... 0.4, 1.2 and 3.5 The long-dash function is zero at these points The graph shows that the solid function is positive (negative) when the slope of the tangent line of the short-dash function is positive... positive (negative) Also, the long-dash function is positive (negative) when the slope of the tangent line of the solid function is positive (negative) Thus, the short-dash function is f, the solid