Answer :
The statement f'(0.4) < f'(-1.5) < f'(-3.1) is true.
f(x) = 1/7x⁷+1/2x⁶-x⁵-15/4x⁴+4/3x³+6x²
f'(x) is the derivative of f(x).
We use two derivative formulas here:
d/dx(xⁿ) = nxⁿ⁻¹
d/dx(cf) = c d/dx(f) .i.e., the derivative of a constant multiple of a function is the constant multiple of the derivative of the function.
Using these,
f'(x) = 7 x 1/7 x⁶ + 6 x 1/2 x⁵ - 5x⁴ 4 x 15/4 x³ + 3 x 4/3 x² + 2 x 6x
= x⁶+ 3x⁵ - 5x⁴ -15x³ + 4x² + 12x
We will find values for f'(x) at x = -3.1, -1.5, 0.4, 3.1, 1.5
f'(-3.1) = (-3.1)⁶+ 3(-3.1)⁵ - 5(-3.1)⁴ -15(-3.1)³ + 4(-3.1)² + 12(-3.1)
= 14.973651
f'(-1.5) = (-1.5)⁶+ 3(-1.5)⁵ - 5(-1.5)⁴ -15(-1.5)³ + 4(-1.5)² + 12(-1.5)
= 4.921875
f'(0.4) = (0.4)⁶+ 3(0.4)⁵ - 5(0.4)⁴ -15(0.4)³ + 4(0.4)² + 12(0.4)
= 4.386816
f'(1.5) = (1.5)⁶+ 3(1.5)⁵ - 5(1.5)⁴ -15(1.5)³ + 4(1.5)² + 12(1.5)
= -14.765625
f'(3.1) = (3.1)⁶+ 3(3.1)⁵ - 5(3.1)⁴ -15(3.1)³ + 4(3.1)² + 12(3.1)
= 913.392711
Hence we get, f'(1.5) < f'(0.4) < f'(-1.5) < f'(-3.1) < f'(3.1)
So the true inequality will be option (d) . f'(0.4) < f'(-1.5) < f'(-3.1)
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