Question

let ( f ) be the function given by ( f(x)=\frac{1}{7} x^{7}+\frac{1}{2} x^{6}-x^{5}-\frac{15}{4} x^{4}+\frac{4}{3} x^{3}+6 x^{2} ). which of the following statements is true?

a ( f^{prime}(-3.1)<f^{prime}(-1.5)<f^{prime}(0.4) )
b ( f^{prime}(-3.1)<f^{prime}(0.4)<f^{prime}(-1.5) )
c ( f^{prime}(-1.5)<f^{prime}(0.4)<f^{prime}(-3.1) )
d ( f^{prime}(0.4)<f^{prime}(-1.5)<f^{prime}(-3.1) )

Explanation:

Step1: Differentiate the function

Using the power rule \((x^n)^\prime = nx^{n - 1}\), we have:
\(f^\prime(x)=x^{6}+3x^{5}-5x^{4}-15x^{3}+4x^{2}+12x\)

Step2: Calculate \(f^\prime(-3.1)\)

Substitute \(x = - 3.1\) into \(f^\prime(x)\):
\(f^\prime(-3.1)=(-3.1)^{6}+3(-3.1)^{5}-5(-3.1)^{4}-15(-3.1)^{3}+4(-3.1)^{2}+12(-3.1)\)
\(f^\prime(-3.1)=887.503681-3\times286.29151 - 5\times92.3521+15\times29.791+4\times9.61-37.2\)
\(f^\prime(-3.1)=887.503681 - 858.87453-461.7605 + 446.865+38.44-37.2\)
\(f^\prime(-3.1)\approx135\)

Step3: Calculate \(f^\prime(-1.5)\)

Substitute \(x=-1.5\) into \(f^\prime(x)\):
\(f^\prime(-1.5)=(-1.5)^{6}+3(-1.5)^{5}-5(-1.5)^{4}-15(-1.5)^{3}+4(-1.5)^{2}+12(-1.5)\)
\(f^\prime(-1.5)=11.390625-3\times7.59375-5\times5.0625 + 15\times3.375+4\times2.25-18\)
\(f^\prime(-1.5)=11.390625-22.78125 - 25.3125+50.625+9 - 18\)
\(f^\prime(-1.5)\approx-15\)

Step4: Calculate \(f^\prime(0.4)\)

Substitute \(x = 0.4\) into \(f^\prime(x)\):
\(f^\prime(0.4)=(0.4)^{6}+3(0.4)^{5}-5(0.4)^{4}-15(0.4)^{3}+4(0.4)^{2}+12(0.4)\)
\(f^\prime(0.4)=0.004096+3\times0.01024-5\times0.0256-15\times0.064 + 4\times0.16+4.8\)
\(f^\prime(0.4)=0.004096 + 0.03072-0.128-0.96+0.64+4.8\)
\(f^\prime(0.4)\approx4.3868\)

Answer:

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