1
WB JEE 2023
+1
-0.25

Let $$f:[1,3] \to R$$ be continuous and be derivable in (1, 3) and $$f'(x) = {[f(x)]^2} + 4\forall x \in (1,3)$$. Then

A
$$f(3) - f(1) = 5$$ holds
B
$$f(3) - f(1) = 5$$ does not hold
C
$$f(3) - f(1) = 3$$ holds
D
$$f(3) - f(1) = 4$$ holds
2
WB JEE 2023
+1
-0.25

f(x) is a differentiable function and given $$f'(2) = 6$$ and $$f'(1) = 4$$, then $$L = \mathop {\lim }\limits_{h \to 0} {{f(2 + 2h + {h^2}) - f(2)} \over {f(1 + h - {h^2}) - f(1)}}$$

A
does not exist
B
equal to $$-3$$
C
equal to 3
D
equal to 3/2
3
WB JEE 2023
+1
-0.25

Let $$f(x) = \left\{ {\matrix{ {x + 1,} & { - 1 \le x \le 0} \cr { - x,} & {0 < x \le 1} \cr } } \right.$$

A
f(x) is discontinuous in [$$-1,1$$] and so has no maximum value or minimum value in [$$-1,1$$].
B
f(x) is continuous in [$$-1,1$$] and so has maximum value and minimum value.
C
f(x) is discontinuous in [$$-1,1$$] but still has the maximum and minimum value.
D
f(x) is bounded in [$$-1,1$$] and does not attain maximum or minimum value.
4
WB JEE 2023
+1
-0.25

Let $$f(x) = [{x^2}]\sin \pi x,x > 0$$. Then

A
f is discontinuous everywhere.
B
f is continuous everywhere.
C
f is continuous at only those points which are perfect squares.
D
f is continuous at only those points which are not perfect squares.
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