Use the derivative function, f ' ( x ) , to determine where the function f ( x ) = − 2 x 2 + 13 x − 8 is increasing.

f is increasing on interval (-infty, 13/4)

Inequality notation: x<13/4

In words: f is increasing on the interval of x that is less than 13/4.

Step-by-step explanation:

f is increasing on interval of x if f' of such interval is positive.

f=-2x^2+13x-8

Differentiate both sides

(f)'=(-2x^2+13x-8)'

Sum and difference rule:

f'=(-2x^2)'+(13x)'-(8)'

Constant multiple rule:

f'=-2(x^2)'+13(x)'-(8)'

Power rule (recall x=x^1):

f'=-2(2x^1)+13(1x^0)-(8)'

Constant rule:

f'=-2(2x^1)+13(1x^0)-(0)

Recall again x^1=x:

f'=-2(2x)+13(1x^0)-(0)

Recall x^0=1:

f'=-2(2x)+13(1×1)-(0)

Associative property of multiplication:

f'=-(2×2)x+13(1×1)-(0)

Performed grouped multiplication:

f'=-(4)x+13(1)-(0)

f'=-4x+13-(0)

Additive identity:

f'=-4x+13

f' is positive when -4x+13>0.

Subtract 13 on both sides:

-4x>-13

Divide both sides by -4:

x<-13/-4

x<13/4

f is increasing on interval (-infty, 13/4)