How do you find and classify all the critical points and then use the second derivative to check your results given #y=x^2+10x-11#?

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Answer 1 · 2016-07-30T02:03:40

Vertex #(-5, -36)#
Y-intercept #(0, -11)#
X-intercepts #(-11, 0) #and #(1, 0)#

Given -

#y=x^2+10x-11#

It is a quadratic equation .
It has only one critical point.
It is the vertex.

#x=(-b)/(2a)=(-10)/(2 xx 1)=-5#

At #x=-5; y= (-5)^2+10(-5)-11#

#y= 25-50-11=25-61=-36#

Vertex is #(-5, -36)#

Derivatives of the function are

#dy/dx=2x+10#
#(d^2y)/(dx^2)=2 > 0#

Its second derivative is greater than zero. The curve is concave upwards.

Its other important points are

Y-intercept

At #x=0; y=0^2+10(0)-11=-11#

At #(0, -11# the curve cuts the Y-axis

X- intercepts

At #y=0; x^2+10x-11=0#

# x^2+11x-x-11=0#
#x( x+11)-1(x+11)=0#
#(x+11)(x-1)=0#
#x+11=0#
#x=-11#

#(-11, 0) # is one of the x- intercept

#x-1=0#
#x=1#

#(1, 0)# is another x-intercept.

At points #(-11, 0) #and #(1, 0)# , the curve cuts the x-axis

Look at the graph