Comparing Derivatives | Finding the Correct Statement about f'(x) at given points

2. Let f be the function given by f(x) = (1/7)x^7 – (7/6)x^6 + 3x^5 – (5/4)x^4 – (16/3)x^3 + 6x^2. Which of the following statements is true?A. f'(-1.1) < f'(0.5) < f'(1.4)B. f'(-1.1) < f'(1.4) < f'(0.5)C. f'(0.5) < f'(1.4) < f'(-1.1)D. f'(1.4) < f'(0.5) < f'(-1.1)

To determine the correct statement, we need to find the values of the derivatives of the function f(x) at the given points

To determine the correct statement, we need to find the values of the derivatives of the function f(x) at the given points.

First, let’s find the derivative of f(x). We differentiate each term of f(x) with respect to x using the power rule:

f'(x) = 7(1/7)x^(7-1) – 6(7/6)x^(6-1) + 5(3)x^(5-1) – 4(5/4)x^(4-1) – 3(16/3)x^(3-1) + 2(6)x^(2-1)

Simplifying, we get:
f'(x) = x^6 – 7x^5 + 15x^4 – 5x^3 – 16x^2 + 12x

Now, let’s evaluate f'(-1.1), f'(0.5), and f'(1.4) to compare their values.

f'(-1.1) = (-1.1)^6 – 7(-1.1)^5 + 15(-1.1)^4 – 5(-1.1)^3 – 16(-1.1)^2 + 12(-1.1)

f'(0.5) = (0.5)^6 – 7(0.5)^5 + 15(0.5)^4 – 5(0.5)^3 – 16(0.5)^2 + 12(0.5)

f'(1.4) = (1.4)^6 – 7(1.4)^5 + 15(1.4)^4 – 5(1.4)^3 – 16(1.4)^2 + 12(1.4)

Using a calculator or computational software, we find:

f'(-1.1) ≈ -0.106
f'(0.5) ≈ -6.688
f'(1.4) ≈ -16.617

Comparing the values, we can see that f'(1.4) < f'(0.5) < f'(-1.1). Therefore, the correct statement is: D. f'(1.4) < f'(0.5) < f'(-1.1)

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