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

The range of f(x)=x^4-4x^3+3x^2 is f(x)>=-3(9+5sqrt(3))+(9(11+6sqrt(3)))/4

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

range

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

f (x) \geq - 3 (9 + 53) + \frac{9 ( 11 + 6 3 )}{4}

Interval Notation

[- 3 (9 + 53) + \frac{9 ( 11 + 6 3 )}{4}, \infty)

Solution steps

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

Domain of

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

Extreme Points of

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

Minimum

(0, 0),

Maximum

(\frac{3 - 3}{2}, - 3 (9 - 53) + \frac{9 ( 11 - 6 3 )}{4}),

Minimum

(\frac{3 + 3}{2}, - 3 (9 + 53) + \frac{9 ( 11 + 6 3 )}{4})

Find the range for the interval

- \infty < x < \infty : - 3 (9 + 53) + \frac{9 ( 11 + 6 3 )}{4} \leq f (x) < \infty

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

f (x) \geq - 3 (9 + 53) + \frac{9 ( 11 + 6 3 )}{4}

9 x^{2} - 6 x - 3 x

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

\frac{7 x ^{2} + 8 x}{8 x ^{2} - 4}

f (x) = \frac{1}{x + 5}

f (x) = \frac{( x - 2 ) ( x + 9 )}{x ^{3}}

What is the range of f(x)=x^4-4x^3+3x^2 ?
The range of f(x)=x^4-4x^3+3x^2 is f(x)>=-3(9+5sqrt(3))+(9(11+6sqrt(3)))/4