Question

Let \(f\) be a differentiable function satisfying \[ f(x)=1-2x+\int_0^x (t-x)f(t)\,dt,\quad x\in\mathbb{R}, \] and let \[ g(x)=\int_0^x \{f(t)+2\}^5(t-4)^6(t+12)^7\,dt. \] If \(p\) and \(q\) are respectively the points of local minima and local maxima of \(g\), then the value of \(|p+q|\) is _______.

Hint:

In integrals defining functions, extrema are found by analysing the sign of the integrand.

Answer 1

Correct Answer: 8

Step 1: Differentiate the functional equation Differentiate once: \[ f'(x)=-2+\int_0^x(-f(t))\,dt \] Differentiate again: \[ f''(x)=-f(x) \]
Step 2: Solve the differential equation \[ f''+f=0 \Rightarrow f(x)=A\cos x+B\sin x \] Using \(f(0)=1\) and \(f'(0)=-2\): \[ A=1,\ B=-2 \] \[ f(x)=\cos x-2\sin x \]
Step 3: Find critical points of \(g(x)\) \[ g'(x)=(f(x)+2)^5(x-4)^6(x+12)^7 \] Critical points: \[ x=4,\ x=-12,\ f(x)+2=0 \] \[ \cos x-2\sin x+2=0 \Rightarrow x=\frac{\pi}{2} \]
Step 4: Nature of extrema \[ (x-4)^6\ \text{even power},\quad (x+12)^7\ \text{odd power} \] Hence: \[ \text{Local minimum at }x=-12,\quad \text{Local maximum at }x=4 \] \[ p=-12,\ q=4 \] Final Answer: \[ \boxed{8} \]