the slope of a curve y=f(x) is given by

Question

dy/dx= (x-1)(x-2)^2(x-3)^3(x-4)^4(x-5)^5

for what value or values of x does y have a local maximum/miniumum? justify your answer.

i tried solving this question but couldn't figure out how to do it. i even asked my old maths school teacher but even she couldn't solve it. could anyone please help me!!! thanks very much

Answer 1

since y'=0 at x=1,2,3,4,5 the max/min can occur there.

But, y' has roots of even multiplicity at x=2 and 4, so those will be inflection points.

So, extrema will occur at x=1,3,5

Why? The slope has to change sign at the extrema. If y' has roots of even multiplicity, it does not change sign there, but only magnitude.

Answer 2

thank you so much for your help