Math

QuestionIdentify if the function f(x)=3x2+6xf(x)=-3 x^{2}+6 x has a max or min value, then find that value.

Studdy Solution

STEP 1

Assumptions1. The given function is a quadratic function, which is a polynomial function of degree. . The general form of a quadratic function is f(x)=ax+bx+cf(x) = ax^ + bx + c, where aa, bb, and cc are constants, and aa is not equal to zero.
3. The given function is f(x)=3x+6xf(x) = -3x^ +6x, which is in the form of f(x)=ax+bxf(x) = ax^ + bx, where a=3a = -3 and b=6b =6.
4. The quadratic function has a maximum value if a<0a <0 and a minimum value if a>0a >0.

STEP 2

First, we need to determine whether the given quadratic function has a maximum value or a minimum value. We can do this by checking the sign of the coefficient aa.

STEP 3

Check the sign of the coefficient aa.
a=3a = -3
Since a<0a <0, the quadratic function has a maximum value.

STEP 4

Next, we need to find the maximum value of the quadratic function. The maximum or minimum value of a quadratic function f(x)=ax2+bx+cf(x) = ax^2 + bx + c is given by the formula b4a-\frac{b}{4a}.

STEP 5

Substitute the values of aa and bb into the formula to find the maximum value.
Maximum value = b4a=4(3)-\frac{b}{4a} = -\frac{}{4(-3)}

STEP 6

Calculate the maximum value.
Maximum value = 64(3)=0.5-\frac{6}{4(-3)} =0.5
The quadratic function has a maximum value of0.5.

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