application of differentiation

Question

Explain with reference to a diagram, how to determine by consideration of the sign of the derivative w.r.t x whether a function of x is increasing or decreasing as x increases

Answer 1

f(x) = 4x

f'(x) = 4>0 therefore, the function is increasing

f(x)=-4x

f'(x) =-4 <0, so the function is decreasing

f(x) =4x^2

f'(x) = 8x which may or may not be positive, depending on the sign of x

f'(3) =8(3) =24>0 is increasing

but

f'(-3) = 8(-3) =-24<0 is decreasing