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Show that the Function F(X) = Xcuberoot3 - 3xsqrt2 + 6x - 100 is Increasing on R - CBSE (Science) Class 12 - Mathematics

ConceptIncreasing and Decreasing Functions

Question

Show that the function f(x) = x^3 - 3x^2 + 6x - 100 is increasing on R

Solution 1

The given function is

f(x) = x3 − 3x2 + 6x −100

∴f'(x) = 3x2 − 6x + 6

=3(x2 − 2x +2)

=3(x2 − 2x + 1) + 3

=3(x−1)2+3

For f(x) to be increasing, we must have f'(x0

Now, 3(x−1)2 ≥ 0  ∀x ∈ R

⇒ 3(x − 1)2 + 3 > 0  ∀x ∈ R

⇒ f'(x) > 0    ∀x ∈ R

Hence, the given function is increasing on R

Solution 2

f(x) = x^3 - 3x^2 + 6x - 100

f'(x) = 3x^2 - 6x + 6

= 3(x^2 -  2x + 1 ) + 3

=3(x+1)^2 + 3 > 0

For all values of x, (x - 1)^2 is always positve

:. f'(x) > 0

So, f (x) is increasing function.

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Solution Show that the Function F(X) = Xcuberoot3 - 3xsqrt2 + 6x - 100 is Increasing on R Concept: Increasing and Decreasing Functions.
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