A calculus problem by vishnu c

By Rolle's Theorem, there are zeroes of $f'(x)$ between a and b, between b and c, and between c and d.

Also, $f'(x)$ has roots at b, c, and d with multiplicities 2, 4, and 6 respectively.

Counting roots with their multiplicities, we have found 15 roots of $f'(x)$ , a polynomial of degree 15, so that there can be no more.

Thus $f'(x)$ has $\boxed{6}$ distinct roots.

well @Otto Bretscher sir, I have found a relatively more analytical method.. $f\left( x \right) =\left( { x-a } \right) ^{ 1 }\left( { x-b } \right) ^{ 3 }\left( { x-c } \right) ^{ 5 }\left( { x-d } \right) ^{ 7 }\\ f^{ / }\left( x \right) =\left( { x-b } \right) ^{ 2 }\left( { x-c } \right) ^{ 4 }\left( { x-d } \right) ^{ 6 }\{ \left( { x-b } \right) ^{ 1 }\left( { x-c } \right) ^{ 1 }\left( { x-d } \right) ^{ 1 }+3\left( { x-a } \right) ^{ 1 }\left( { x-c } \right) ^{ 1 }\left( { x-d } \right) ^{ 1 }+5\left( { x-a } \right) ^{ 1 }\left( { x-b } \right) ^{ 1 }\left( { x-d } \right) ^{ 1 }+7\left( { x-a } \right) ^{ 1 }\left( { x-b } \right) ^{ 1 }\left( { x-d } \right) ^{ 1 }\} \\ now\quad in\quad brackets\quad is\quad a\quad cubic\quad equation-\\ let\quad the\quad expression\quad in\quad \{ \} \quad brackets\quad be\quad g\left( x \right) ,\quad for\quad a<b<c<d-\\ g\left( a \right) =negative\\ g\left( b \right) =positive\\ g\left( c \right) =negative\\ g\left( d \right) =positive\\ \\ clearly\quad g(x)\quad has\quad 3\quad real\quad and\quad disinct\quad roots\quad by\quad drawing\quad a\quad rough\quad graph\quad and\quad also\quad in\quad 1st\quad derivative\quad of\quad f(x)\quad we\quad have\quad 3\quad more\quad solutions,\\ therefore\quad total\quad no.\quad of\quad solutions\quad are-\\ 3+3=6$

Draw the graph of f(x) and consider these points.

Between a and b, there's a minimum.

Then, at b, there are two roots of f'(x), but we shall count it only as one distinct root.

Then, between b and c, there's a maximum. So, that's another root.

Then, at c, you have 4 roots of the derivative that shall only be counted as one root.

Between c and d, you have one minimum.

Again, at d, you have 6 roots that shall only be counted as one distinct root of the derivative.

Thus you have 6 distinct roots.