1
WB JEE 2018
+1
-0.25
Let f : [a, b] $$\to$$ R be differentiable on [a, b] and k $$\in$$ R. Let f(a) = 0 = f(b). Also let J(x) = f'(x) + kf(x). Then
A
J(x) > 0 for all x $$\in$$ [a, b]
B
J(x) < 0 for all x $$\in$$ [a, b]
C
J(x) = 0 has at least one root in (a, b)
D
J(x) = 0 through (a, b)
2
WB JEE 2018
+1
-0.25
Let $$f(x) = 3{x^{10}} - 7{x^8} + 5{x^6} - 21{x^3} + 3{x^2} - 7$$.

Then $$\mathop {\lim }\limits_{h \to 0} {{f(1 - h) - f(1)} \over {{h^3} + 3h}}$$
A
does not exist
B
is $${{50} \over 3}$$
C
is $${{53} \over 3}$$
D
is $${{22} \over 3}$$
3
WB JEE 2018
+1
-0.25
Let f : [a, b] $$\to$$ R be such that f is differentiable in (a, b), f is continuous at x = a and x = b and moreover f(a) = 0 = f(b). Then
A
there exists at least one point c in (a, b) such that f'(c) = f(c)
B
f'(x) = f(x) does not hold at any point in (a, b)
C
at every point of (a, b), f'(x) > f(x)
D
at every point of (a, b), f'(x) < f(x)
4
WB JEE 2018
+1
-0.25
Let f : R $$\to$$ R be a twice continuously differentiable function such that f(0) = f(1) = f'(0) = 0. Then
A
f''(0) = 0
B
f''(c) = 0 for some c$$\in$$R
C
if c $$\ne$$ 0, then f''(c) $$\ne$$ 0
D
f'(x) > 0 for all x $$\ne$$ 0
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