how to find the equation of maximum slope of a function?

hm010

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Im having some trouble with a question, where i need to differentiate a function, and then find the equation with for the tangent line with maximum slope. What steps do i need to follow to find this?
 
Im having some trouble with a question, where i need to differentiate a function, and then find the equation with for the tangent line with maximum slope. What steps do i need to follow to find this?
Is it function of single variable [e.g. y = f(x)] or multiple-variable [e.g. z = f(x,y)]?
 
Im having some trouble with a question, where i need to differentiate a function, and then find the equation with for the tangent line with maximum slope. What steps do i need to follow to find this?

Assuming that you have a function of a single valued function, y= f(x), the first thing you would do is take the derivative of y, y'= df/dx which gives the slope of the tangent line at any x. Then look for the maximum slope. You have probably learned that the (local) maximum value of a differentiable function occurs where its derivative is 0. So if f is twice differentiable, differentiate it again! The point where f''(x)= 0 must be either a (local) maximum or minimum of f'(x). (Check to be sure it is a maximum and not a minimum.). Once you have found that value of x, \(\displaystyle x_0\), the tangent line to y= f(x) there is \(\displaystyle y= f'(x_0)(x- x_0)+ f(x_0)\).
 
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