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Local maxima and minima of a function defined by an integral

This type of question, asked 4 times: 27 Jun 2022, 31 Jan 2024, 22 Jan 2025, 28 Jan 2026 S2. All 4 questions below, oldest first.

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  1. Question 1

    27 Jun 2022

    157
    If mm and nn respectively are the number of local maximum and local minimum points of the function f(x)=∫0x2t2−5t+42+etdtf(x) =\int_{0}^{x^{2}} \frac{t^{2}- 5t + 4}{2 +e^{t}}dt, then the ordered pair (m,n)(m,n) is equal to

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  2. Question 2

    31 Jan 2024

    82
    Let S=(−1,∞)S = ( - 1,\infty) and f:S→Rf:S \rightarrow R be defined as f(x)=∫−1x(et−1)11(2t−1)5(t−2)7(t−3)12(2t−10)61dtf(x) =\int_{- 1}^{x} \left( e^{t}- 1 \right)^{11}(2t - 1)^{5}(t - 2)^{7}(t - 3)^{12}(2t - 10)^{61}dt. Let p=p = Sum of square of the values of xx, where f(x)f(x) attains local maxima on S. and q=q = Sum of the values of xx, where f(x)f(x) attains local minima on SS. Then, the value of p2+2qp^{2}+ 2q is

  3. Question 3

    22 Jan 2025

    133
    Let f(x)=∫0x2t2−8t+15etdt,x∈Rf(x) =\int_{0}^{x^{2}} \frac{t^{2}- 8t + 15}{e^{t}}dt,x \in R. Then the numbers of local maximum and local minimum points of ff, respectively, are :

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  4. Question 4

    28 Jan 2026 S2

    73
    Let ff be a differentiable function satisfying
    f(x)=1−2x+∫0xe(x−t)f(t)dt,x∈R and let f(x) = 1 - 2x + \int_{0}^{x} e^{(x - t)}f(t)dt,x \in R\text{~and let~}
    g(x)=∫0x(f(t)+2)15(t−4)6(t+12)17dt,x∈Rg(x) = \int_{0}^{x} (f(t) + 2)^{15}(t - 4)^{6}(t + 12)^{17}dt,x \in R
    If pp and qq are respectively the points of local minima and local maxima of g , then the value of ∣p+q∣|p + q| is equal to ............ .