Home > Calculus > Identifying Turning Points (Local Extrema) for a Function

What are the local extrema, if any, of #f(x)= 2x+15x^(2/15)#?

Answer 1

William Abdalla

Local maximum of 13 at 1 and local minimum of 0 at 0.

Domain of #f# is #RR#

#f'(x) = 2+2x^(-13/15) = (2x^(13/15)+2)/x^(13/15)#

#f'(x) = 0# at #x = -1# and #f'(x)# does not exist at #x = 0#.

Both #-1# and #9# are in the domain of #f#, so they are both critical numbers.

First Derivative Test:

On #(-oo,-1)#, #f'(x) > 0# (for example at #x = -2^15#) On #(-1,0)#, #f'(x) < 0# (for example at #x = -1/2^15#)

Therefore #f(-1) = 13# is a local maximum.

On #(0,oo)#, #f'(x) >0# (use any large positive #x#)

So #f(0) = 0# is a local minimum.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Adam Adkins

The function (f(x) = 2x + 15x^{\frac{2}{15}}) has a local minimum at (x = -\frac{3}{2}) and no local maximum.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.