Continuity and Differentiability
𝑓 ′ (𝑥) = (𝑒 (𝑓(𝑥)−𝑔(𝑥)) )𝑔′(𝑥) for all 𝑥 ∈ ℝ,
and 𝑓(1) = 𝑔(2) = 1, then which of the following statement(s) is (are) TRUE?
𝑓 (𝑥) = 1 − 2𝑥 + 𝑒𝑥−𝑡 𝑓 (𝑡) 𝑑t
for all 𝑥 ∈ [0, ∞). Then, which of the following statement(s) is (are) TRUE?
If then which of the following statement(s) is (are) TRUE?
Then, the value of loge(f(4)) is _____.
(ii) where the inverse trigonometric function tan–1 x assumes values in ,
(iii) f3(x) = [sin(loge(x + 2)), where, for t ∈ ℝ, [t] denotes the greatest integer less than or equal to t,
|P.||The function f1 is||1.||NOT continuous at x = 0|
|Q.||The function f2 is||2.||continuous at x = 0 and NOT differentiable at x = 0|
|R||The function f3 is||3.||differentiable at x = 0 and its derivative is NOT continuous at x = 0|
|S.||The function f4 is||4.||differentiable at x = 0 and its derivative is continuous at x = 0|
The correct option is:
and h(x) = e|x| for all x ∈ . Let (f ° h) (x) denote f(h(x)) and (h ° f) (x) denote h (f(x)). Then which of the following is (are) true ?
|x = −1||x = 0||x = 2|
The number of points at which h(x) is not differentiable is
|List I||List II|
|P.||f4 is||1.||onto but not one-one|
|Q.||f3 is||2.||neither continuous nor one-one|
|R.||f20f1 is||3.||differentiable but not one-one|
|S.||f2 is||4.||continuous and one-one|
For every integer n, let and real numbers. Let function be given by
, for all integer n.
If f is continuous, then which of the following hold (s) for all n?
Let g(x) =, m and n are integers,, n > 0, and let p be the left hand derivative of at x = 1. If, then
Match the functions in Column I with the properties given in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.
|(A)||x |x|||(p)||Continuous in (−1, 1)|
|(B)||(q)||Differentiable in (−1, 1)|
|(C)||x + [x]||(r)||Strictly increasing in (−1, 1)|
|(D)|||x − 1| + |x + 1|||(s)||Not differentiable at least at one point in (−1, 1)|
If f(x) is a twice differentiable function such that f(a) = 0, f(b) = 2, f(c) = − 1, f(d) = 2 and f(e) = 0, where a < b < c < d < e, then the minimum number of zeroes of the function g(x) = (f′(x))2 + f′′(x) f(x) in the interval [a, e] is
If |f(x1) – f(x2)| < (x1 – x2)2, for all x1, x2 ∈R, what is the equation of tangent to the curve y = f(x) at point (1, 2)?
If f(x – y) = f(x). g(y) – f(y). g(x) and g(x – y) = g(x). g(y) + f(x). f(y) for all x, y ∈ R and the right-hand derivative at x = 0 exists for f(x), then what is the derivative of g(x) at x = 0?
f(x) is a differentiable function and g(x) is a double differentiable function such that
|f(x)| ≤ 1 and f′(x) = g(x). If f2(0) + g2(0) = 9, prove that there exists some c ∈ (– 3, 3)
such that g(c) g′′(c)< 0.
If f(x) is differentiable at x = 0 and | c |<, then find the value of a and prove that 64b2 = (4 – c2).