If \[ f(x)=1-2x+\int_{0}^{x} e^{x-t} f(t)\,dt \] and \[ g(x)=\int_{0}^{x} (f(t)+2)^{11}(t+12)^{17}(t-4)^4\,dt, \] If local minima and local maxima of $g(x)$ occur at $x=p$ and $x=q$ respectively, find $|p|+q$.

Question

If domain of $f(x) = \sqrt{\log_{0.6} \frac{2x - 5}{x^2 - 4}}$ is $(-\infty, a] \cup \{b\} \cup [c, d) \cup (e, \infty)$ then the value of $(a + b + c + d + e)$ is :

Answer 1

To solve the problem, we first analyze the given functions f(x) and g(x), and then determine the points of local minimum and maximum of g(x).

Step 1: Understanding the function f(x)

The function f(x) is defined by the integral equation:

f(x) = 1 − 2x + ∫₀ˣ e^(x−t) f(t) dt

This is an integral equation. Solving it using standard methods (differentiation and verification), we obtain:

f(x) = eˣ (1 − 2x)

Step 2: Expression for g(x)

The function g(x) is given by:

g(x) = ∫₀ˣ (f(t) + 2)¹¹ (t + 12)¹⁷ (t − 4)⁴ dt

Local maxima and minima of g(x) occur where g′(x) = 0.

Differentiating g(x):

g′(x) = (f(x) + 2)¹¹ (x + 12)¹⁷ (x − 4)⁴

Step 3: Critical points

Setting g′(x) = 0, we find the zeros of the expression:

x + 12 = 0 ⇒ x = −12
x − 4 = 0 ⇒ x = 4

These points correspond to local extrema because the remaining factors do not change sign at these values.

Hence:

Local minimum at p = −12
Local maximum at q = 4

Step 4: Required value

|p| + q = 12 + 3 = 15

Final Answer:

15

Answer 2