Home > Calculus > Interpreting the Sign of the First Derivative (Increasing and Decreasing Functions)

Is #f(x)=(x^3-5x^2-x+2)/(2x-1)# increasing or decreasing at #x=0#?

Answer 1

Emilia Aguilar

decreasing

#d/dx f(x) = ((3x^2-5x-1)*(2x-1) - (2(x^3-5x-x+2)))/(2x-1)^2#

if #x=0#

#d/dx f(0) = (1-2)/(1) = -1#

so it's decreasing on #x=0#

It's even displayed on the graph below.

graph{[-10, 10, -5, 5]}/(2x-1) (x^3-5x^2-x+2)

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Answer 2

Emilia Aguilar

To determine whether f(x) = (x^3 - 5x^2 - x + 2) / (2x - 1) is increasing or decreasing at x = 0, we can analyze the sign of the derivative at x = 0.

Compute the derivative of f(x) using the quotient rule:

f'(x) = [(2x - 1)(3x^2 - 10x - 1) - (x^3 - 5x^2 - x + 2)(2)] / (2x - 1)^2

Evaluate the derivative at x = 0:

f'(0) = [(0 - 1)(0 - 0 - 1) - (0 - 0 - 0 + 2)(2)] / (0 - 1)^2 = [(−1)(−1) − (2)(2)] / 1 = [1 - 4] / 1 = -3

Since f'(0) is negative (-3), the function f(x) is decreasing at x = 0.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.