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For each function, the point given is the maximum or minimum. Use the difference quotient to verify that the slope of the tangent at this point is zero.

a) f(x) = 0.5x^2 + 6x + 7.5; (-6, -10.5)

Difference quotient is

f(a + h) - f(a)/h

m= f(a+h) - f(a)/h
= f(-6+h) - f(-6)/h
= 0.5(-6+h)^2 + 6(-6+h)+ 7 - (-10.5)/h

What do I do next?

= 0.5(-6+h)^2 + 6(-6+h)+ 7.5 - (-10.5)/h *

just go ahead and work it out ...

m = [ .5(36 - 12h + h^2) - 36 + 6h + 7.5 + 10.5]/h
= [ 18 - 6h + h^2/2 - 36 + 6h + 7.5 + 10.5]/h
= (h^2/2)/ h
= h/2

now as h ---> 0 , m = 0

how did you get h^2/2?

somebody? :|

.5 is the same as 1/2, so
.5(h^2) = (1/2)h^2 = h^2/2

Oh wow thanks~

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