please help!!!! f is given by f(x)=1/7x7+1/2x6-x5-15/4x4+4/3x3+6x2. What is true?

Question

f is given by f(x)=1/7x⁷+1/2x⁶-x⁵-15/4x⁴+4/3x³+6x². What is true?

A)f')-3.1) < f'(-1.5) < f'(0.4)

B) f'(3.1) < f'(0.4) < f'(-1.5)

C) f'(-1.5) < f'(0.4) < f'(-3.1)

D) f'(0.4) < f'(1.5) < f'(-3.1)

Answer 1

By differentiating f(x) we get

f'(x) = x⁶ + 3x⁵ - 5x⁴ -15x³ + 4x² +12x

At this point you could simply evaluate f' at the values

-3.1, -1.5, 0.4, 1.5, 3.1 and compare them.

However, I am pretty sure that your teacher wants you to convert f'(x) to a factored form.

Perhaps because sums are less numerically stable than products?

Did you talk about that?

In any case here is how to convert f'(x) to factored form:

f'(x) = x(x⁵ + 3x⁴ - 5x³ -15x² + 4x +12)

= x(x⁴(x+3) - 5x²(x+3) + 4(x+3))

= x(x+3)(x⁴ - 5x² + 4)

= x(x+3)(x² - 4)(x² - 1)

= x(x+3)(x-2)(x+2)(x-1)(x+1)

From the factored form we see that the roots of f'(x) are

-3, -2, -1, 0, 1, -2

If we evaluate f'(x) at -3.1, -1.5, 0.4, 1.5, 3.1

we see that none of the four identities are true.

I am sorry, I do not see what was intended by this exercise,

except for the practice of factoring.