Application of DerivativeHard
Question
Which of the following statements are true :
Options
A.|tan-1 x - tan-1 y| |x – y| , where x, y are real numbers.
B.The function x100 + sin x - 1 is strictly increasing in [0, 1]
C.If a, b, c are is A.P, then at least one root of the equation 3ax2 - 4bx + c = 0 is positive
D.Curve y2 = 4ax and y = e-x/2a are orthogonal curves.
Solution
(A) let f(x) = tan-1x
f′(x) =
∴ |f′(c)| =
< 1
< 1
(B) Let f(x) = x100 + sin x - 1
f′(x) = 100x99 + cosx > 0, x ∈ [0, 1]
⇒ f(x) is increasing.
(C) Suppose f(x) = ax3 - 2bx2 + cx, then clearly f(0) = 0
and f(1) = a - 2b + c = 0,
∵ f(0) = f(1)
∴ By Rolle′s theorem f′(x) = 3ax2 - 4bx + c = 0
for atleast one x in (0, 1) which is positive
(D) y2 = 4ax ⇒
y =
=
y
Product of slopes =
= - 1
f′(x) =
∴ |f′(c)| =
< 1 < 1(B) Let f(x) = x100 + sin x - 1
f′(x) = 100x99 + cosx > 0, x ∈ [0, 1]
⇒ f(x) is increasing.
(C) Suppose f(x) = ax3 - 2bx2 + cx, then clearly f(0) = 0
and f(1) = a - 2b + c = 0,
∵ f(0) = f(1)
∴ By Rolle′s theorem f′(x) = 3ax2 - 4bx + c = 0
for atleast one x in (0, 1) which is positive
(D) y2 = 4ax ⇒
y =
=
yProduct of slopes =
= - 1Create a free account to view solution
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