slope of tangent line to curve takes largest value?

Question

I'm confused on this question..

For what values(s) of x does the slope of the tangent line to the curve y= -x^3 + 3x^2 +1 take on its largest value? Justify your answer

thanks

Answer 1

y = -x^3 + 3x^2 +1

The slope is the derivative of the function:
dy/dx = -3x^2 + 6x

To get the maximum (or minimum) you want to set it's derivative to 0
d^2y/d^2x = -6x + 6 = 0 and x = 1

Tp justify this, take another derivative (which will be the second derivative of the slope).
d^3y/d^3x = -6x = -6
Therefore the slope at x=1 is a maximum

Answer 2

Let's find the equation of the tangent to the curve:

dy/dx = -3x^2 + 6x

Let, m = -3x^2 + 6x

We have to find for what values of x is dy/dx maximum.

So, take the derivative of m with respect to x.

Hence:

dm/dx = -6x + 6

Equate it to 0.

-6x + 6 = 0
6x = 6
x = 1

Hence, the maximum value of the slope of the tangent to the curve occurs at x = 1.

Answer 3

Since it's a cubic graph switched around the y axis, the slope of the tangent line when x approaches positive or negative infinity would be the lowest.

Being as the graph has and inflection point at (1,3), that is where the slope will be the largest, since the slope of the graph is mostly negative. When the graph shifts from upwards to downwards concavity, it will have its greatest slope there.

So, when x=1, the graph will have its greatest slope.

Answer 4

y= -x^3 + 3x^2 +1
Slope of the tangent = dy/dx

dy/dx = -3x^2 + 6x
Call this u(x)

u(x) = -3x^2 + 6x
u'(x) = -6x + 6
u"(x) = -6 < 0 This will be a maximum

-6x + 6 = 0
6x = 6
x = 1

u'(1) = 0 ---- This means that at x=1, u is either a maximum, minimuim or a point of inflexion.
u"(1) < 0 ---- This means that at x=1, u is a maximum

Answer 5

x=1 i used graphmatica...
my try is...

using leading coefficient test...
ax^n (where it is the one with the highest exponent)
a<0 and n is odd meaning the graph rises to the left and falls to the right(basing from the origin)

sketching the graph we can see that a curve like motion from -1 and 3 we must get its midpoint to get the point with the highest slope

(-1+3)/2=2/2=1

x=1

i think

Answer 6

well all i know is.. the slope of the tangent line is derivative.. so.. find the derivative? or the biggest derivative?


so like.. 3x^2+6x and find the value of of x for that?