Question

The graph of the derivative $f^{\prime}$ 'of a continuous function $f$ is shown with $f(0)=0$. If
(i) $f$ is monotomic increasing in the interval $[ a , b ) \cup( c , d ) \cup( e , f ]$ and decreasing in $( p , q ) \cup( r , s )$.
(ii) $f$ has a local minima at $x = x _1$ and $x = x _2$.
(iii) $f$ is concave up in $(l, m ) \cup( n , t ]$
(iv) $f$ has inflection point at $x = k$
(v) number of critical points of $y=f(x)$ is ' $w$ '.
Find the value of $( a + b + c + d + e )+( p + q + r + s )+( l + m + n )+\left( x _1+ x _2\right)+( k + w )$.

Answer 1

(i) increasing on $[0,2),(4,6),(8,9]$ as $f ^{\prime}( x )>0$ and decreasing on $(2,4),(6,8)$ as $f ^{\prime}( x )<0$
(ii) local minima at $x=4,8$
(iii) $CU$ on $(3,6),(6,9]$
(iv) $k =3$
(v) $ w=4$ i.e. $(x=2,4,6$ and 8$)$
hence
$( a + b + c + d + e )+( p + q + r + s )+( l + m + n )+\left( x _1+ x _2\right)+( k + w ) $
$=(0+2+4+6+8)+(2+4+6+8)+(3+6+6)+(4+8)+(3+4) $
$=(20)+(20)+(15)+(12)+(7)=74 $