MHTCET 17 Apr 2026 Shift 1 Paper

Concept:The derivative of an integral function is the integrand. So, the sign of $f'(x)$ tells whether $f(x)$ increases or decreases.Explanation:Given $f(x)=\int \frac{x+3}{x^{2}-9x+20}\,dx$.Differentiating both sides, we get $f'(x)=\frac{x+3}{x^{2}-9x+20}$.Factorize the denominator: $x^{2}-9x+20=(x-4)(x-5)$.So, $f'(x)=\frac{x+3}{(x-4)(x-5)}$.The function is not defined at $x=4$ and $x=5$.The critical points are $x=-3,4,5$.Check the sign of $f'(x)$ on each interval.For $x<-3$: $x+3<0$ and $(x-4)(x-5)>0$, hence $f'(x)<0$. So, $f(x)$ decreases on $(-\infty,-3]$.For $-3 < x<4$: $x+3>0$ and $(x-4)(x-5)>0$, hence $f'(x)>0$. So, $f(x)$ increases on $(-3,4)$.For $4 < x<5$: $x+3>0$ and $(x-4)(x-5)<0$, hence $f'(x)<0$. So, $f(x)$ decreases on $(4,5)$.For $x>5$: $x+3>0$ and $(x-4)(x-5)>0$, hence $f'(x)>0$. So, $f(x)$ increases on $(5,\infty)$.Therefore, $f(x)$ decreases on $(-\infty,-3] \cup (4,5)$.Answer:Option C: decreases on $(-\infty,-3] \cup (4,5)$.