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Theorems on Integration

If the functi...

Question

If the function f(x)= x³-3ax²+b is strictly increasing derivative for x > 0, then which of the following is always true?

A

a can take any real value

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B

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C

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D

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Solution

The correct option is D
f(x) = x³ -3ax² + b

f'(x) = 3x² - 6ax

for f(x) to be increasing f'(x) > 0.

3x² - 6ax > 0

3x (x - 2a) > 0

(x - 2a) > 0 [There fore x > 0]

x > 2a

So, a ≥0

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Similar questions

If the function f(x)= x³-3ax²+b is strictly increasing derivative for x > 0, then which of the following is always true?

Q. Without using the derivative, show that the function f (x) = | x | is
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0).

f : [0, \infty) \to R

be a strictly increasing function such that the functions

g (x) = f (x) - 3 x

h (x) = f (x) - x^{3}

are both strictly increasing function. Then the function

F (x) = f (x) - x^{2} - x

f (x)

is a differentiable function and

f (x) - x^{3}

is strictly increasing, then

f (x) + x - x^{2}

Q. Determine for what values of

x

f (x) = x^{3} + \frac{1}{x^{3}} (x \neq 0)

is strictly increasing or strictly decreasing.

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