Find the critical points of the function and use the First Derivative Test to de

Question

Find the critical points of the function and use the First Derivative Test to determine whether the critical point is a local minimum or maximum (or neither). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) y = - x^2 + 8x + 2 local minimum c = _____ local maximum c = _____ Determine the intervals on which the function is increasing or decreasing. (Enter your answers using interval notation. Enter EMPTY or emptyset for the empty set.) increasing ______ decreasing _____

Explanation / Answer

given f(x)=y =-x2+8x+2

differentiate with respect to x

f '(x)=dy/dx =-2x+8

f '(c)=-2c+8

for critical point f '(c)=0

-2c+8=0

=>c=4

when c<4, f '(c)>0

when c>4, f '(c)<0

so f '(c) changes from positive to negative at c =4

so critical point is a local maximum

f '(c)>0 when c<4,

so function is increasing in (-,4)

f '(c)<0 when c>4

so function is decreasing in (4,)

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