State the actual number of real zeros of each function fx x

State the actual number of real zeros of each function. f(x) = -x^4 + 4x^2 + 2x - 2 # Real Zeros: 3 # # Real Zeros: 2 # Real Zeros: 1 # Real Zeros: 0 f(x) = x^3 - 3x^2 + 5 # Real Zeros: 1 # Real Zeros: 0 # Real Zeros: 2 # Real Zeros: 3 Approximate the real zeros of each function to the nearest tenth. f(x) = x^4 - 3x^2 -2x - 1 # Real Zeros: x = -2.1, x = 1.6 Real Zeros: x = -1, x = 1.4 Real Zeros: x = -1.4, x = 2.1 Real Zeros: x = -1 f(x) = -x^3 + 3x^2 - 4 Real Zeros: x = 3 Real Zeros: x = -4 Real Zeros: x = -1, x = 2 Real Zeros: x = 0, x = -4 f(x) = -x^4 + x^3 +x^2 - 4 Real Zeros: x = 0 Real Zeros: x = 1.2 Real Zeros: None Real Zeros: x = -0.4, x = 1.2 f(x) = x^3 - 4x^2 + 2 Real Zeros: x = -0.7, x = 0.8, x = 3.9 Real Zeros: x = -0.7, x = 2, x = 3.9 Approximate the relative minimum and relative maxima of each function to the nearest tenth. f(x) = x^4 - 4x^2 + 3x + 5 Minima: (-1.6, -3.5) Maxima: (0.4,5.6) Minima: (-0.7, 3.8), (0,7,3.8) Maxima: (0,4) Minima: None Maxima: (0,2,2.2) Minima: (-1.6, -3.5), (1.2, 4.9) Maxima: (0.4,5.6) f(x) = -x^2 + 3x^2 - 3 Minima: (0, 2) Maxima: (1.3, 3.2) Minima: (2,1) Maxima: (0,-3) Minima: (0, 3) Maxima: (1.3,4.2) Minima: (0, -3) Maxima: (2, 1) Solve each system. 5x - 7y - 5z = 1 5x + 5y + 4z = 25 -5x + y - 2z = -13 (3,2,0) Infinitely many solutions (2, 4, 3) (4, 2, 3) x - y - 6z = -14 -2x - 5y = 15 6x + 4y + z = 14 (4,2,-6) (5,-5,4) (-2, 0, 6) (-6, 4,2)

Solution

51. -x⁴ + 4x² + 2x - 2

To find the real zeroes, we have to check the sign change of the terms.

Here, we can notice that sign change is only twice.

First from first to second term.

Second from third to fourth term.

Hence, real zeroes are 2.