RankLab
Back to Concepts
MathsMediumClass 12

Intermediate Value Theorem

Limits & Continuity

9

JEE Qs

8%

Hard

45

min

Always explicitly state and verify the continuity condition on the closed interval before applying IVT, especially for proving existence of roots.

🧮 Key Formulas

If f is a continuous function on a closed interval [a, b], and k is any real number between f(a) and f(b) (i.e., f(a) < k < f(b) or f(b) < k < f(a)), then there exists at least one c in the open interval (a, b) such that f(c) = k.
Special case for roots: If f is continuous on [a, b] and f(a) and f(b) have opposite signs (i.e., f(a) * f(b) < 0), then there exists at least one c in (a, b) such that f(c) = 0 (a root of f(x) = 0).

✅ Key Points for JEE

  • 1The Intermediate Value Theorem (IVT) crucially relies on the function being continuous over a closed interval [a, b]. If continuity is not met, the theorem does not apply.
  • 2The theorem is an existence theorem; it guarantees that at least one 'c' exists but does not provide a method to find 'c' or assert its uniqueness.
  • 3A primary application is proving the existence of roots for equations: if a continuous function changes sign over an interval, it must cross the x-axis at least once within that interval.
  • 4IVT implies that a continuous function takes on every value between its values at the endpoints of an interval, meaning its graph cannot 'jump' over any y-value.

⚠️ Common Mistakes

  • ✕Applying the theorem without verifying the continuity of the function on the given closed interval [a, b].
  • ✕Assuming that the 'c' guaranteed by the theorem is unique, when in fact there can be multiple such values.
  • ✕Misinterpreting the interval for 'c'; it exists in the *open* interval (a, b) unless f(a) = k or f(b) = k.

📝 Practice Questions

See all

Q7. (2x2−3x+5)(3x−1) 2 limx→∞ is equal to : (3x2+5x+4)√(3x+2)x (1) 2 (2) 2e √3e √3 (3) 2 (4) 2e 3√e 3

2025·MCQEasy

Q19.Consider the region R = {(x, y) : x ≤y ≤9 −113 x2, x ≥0}. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R , is: (1) 730 (2) 625 119 111 (3) 821 (4) 567 123 121

2025·MCQHard

Q11.If limx→∞(( 1−e ) ( e − 1+x )) = α, then the value of 1+loge α equals : (1) e−1 (2) e2 (3) e−2 (4) e

2025·MCQHard

Q7. x2 {sin (k1 + 1)x + sin (k2 −1)x}, x < 0 ⎧ If the function f(x) = 4, x = 0 is continuous at x = 0, then k21 + k22 is ⎨ 2 2+k1x x > 0 x loge ( 2+k2x ), ⎩ equal to (1) 20 (2) 5 (3) 8 (4) 10

2025·Multi conceptHard

Q14. IfI(m, n) = ∫10 xm−1(1 −x)n−1dx, m, (1) I(19, 27) (2) I(9, 1) (3) I(1, 13) (4) I(9, 13)

2025·MCQHard

Q9. Let [x] denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function f(x) = [x] + |x −2|, −2 < x < 3, is not continuous and not differentiable. Then m + n is equal to : (1) 6 (2) 8 (3) 9 (4) 7

2025·MCQMedium

NCERT Chapters

  • Class 12 Maths Ch 5: Continuity and Differentiability