Describe the graph of $f$ if the graph of its integral $g(t) = \int_{0}^{t} f(s) ds $ is:
graphic of g
graphic of f
I analyze the derivative and the sign of the derivative and try to find concave point, and see what happen ion the points $0,a,b,c,d,1$ but i stuck, some help please.
I think continuity of f should not hold. Applying Newton-Leibniz theorem, we get g'(t)=f(t), or, g"(t)=f'(t). Note g"(t) is continuously varying, so f'(t) should not be a constant. In other words, do not use straight lines to join points (0,a,b,c,intermediate-point(say e),d). b,e are points of inflection with a vertical tangent to the curve, so, g'(b),g'(e) are undefined,or, f(b),f(e) are undefined. So, do not draw pointed edges at b, for then, continuity of f holds, which should not be, because continuity holds iff LHL at b=RHL at b=f(b)(the same with e as well), which cannot be, because f(b),f(e) are undefined.