What is the range of f(x)=x^4-2x^3+x-1 ?

The range of f(x)=x^4-2x^3+x-1 is f(x)>= ((1-sqrt(3))^4)/(16)+sqrt(3)-3

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range of f(x)=x^4-2x^3+x-1

range

f (x) = x^{4} - 2 x^{3} + x - 1

f (x) \geq \frac{( 1 - 3 ) ^{4}}{16} + 3 - 3

Interval Notation

[\frac{( 1 - 3 ) ^{4}}{16} + 3 - 3, \infty)

Solution steps

Find the minimum and maximum value in each defined interval and unite the results

Domain of

x^{4} - 2 x^{3} + x - 1 : - \infty < x < \infty

Extreme Points of

x^{4} - 2 x^{3} + x - 1 :

Minimum

(\frac{1 - 3}{2}, \frac{( 1 - 3 ) ^{4}}{16} + 3 - 3),

Maximum

(\frac{1}{2}, - \frac{11}{16}),

Minimum

(\frac{1 + 3}{2}, \frac{( 1 + 3 ) ^{4}}{16} - 3 - 3)

Find the range for the interval

- \infty < x < \infty : \frac{( 1 - 3 ) ^{4}}{16} + 3 - 3 \leq f (x) < \infty

Combine the ranges of all domain intervals to obtain the function range

f (x) \geq \frac{( 1 - 3 ) ^{4}}{16} + 3 - 3

x^{2} - 2 x - 15

f (x) = \frac{\frac{x}{x + 2}}{\frac{x}{x + 2} + 2}

h (t) = \frac{t ^{2} - 4 t}{t ^{4} - 256}

3 x + 12 y = 24

f (x) = 3 x - 4

What is the range of f(x)=x^4-2x^3+x-1 ?
The range of f(x)=x^4-2x^3+x-1 is f(x)>= ((1-sqrt(3))^4)/(16)+sqrt(3)-3