How do I find 2 unknown variables of f(x) when given the tangent line to a function at a point?

Find a and b if the tangent line to the function #f(x)=x^4#+a#x^2#+b at x=1 is y=2x+3

How do I find a and b in this equation?

1 Answer
Jun 6, 2018

Take the derivative of #f(x)# to find your slope at #x=1#, and solve for #a# then backsolve for #b#. you will find that #a=-1# and #b=5#

Explanation:

So we know by the definition of tangent that the tangent line should be equal to the function at #x=1#. We also know that the slope of the tangent line should be equal to the derivative of the function at the same point. Let's find the derivative of #f(x)# and plug in #x=1#:

#f(x)=x^4+ax^2+b#
#f'(x)=4x^3+2ax#

#f'(1)=4(1)^3+2a(1)#
#f'(1)=4+2a#

If the equation for the tangent line is #color(red)(y=2x+3#, then the slope of the line (which is equal to #f'(1)#) is 2.

#f'(1)=2=4+2a#

#-2=2a#

#color(blue)(a=-1)#

Now that we know #a#, we can go back and solve for #b#:

#f(1)=(1)^4+color(blue)((-1))(1)^2+b=color(red)(2(1)+3)#

#f(1)=1-1+b=color(red)(5)#

#color(green)(b=5)#