Question

If f(x) is strictly concave up, then
f '(a) < (f(b) – f(a))/(b – a)
for a < b.

true or false?

Answer 1

TRUE
lets check
\[f(x)=x^3\]is strictly increasing
if we take a=0,and a<b let b=1
\[f^'(0)=3(0)^2=0\]
the average gradient between a and b
\[=\frac{ f(b)-f(a) }{ b-a }\] is\[\frac{ 1^3-0^3 }{ 1-0 }=1\]
\[f^'(a)<average, gradient\],if we try again for x=1 and b=2
\[f^'(1)=3\]\[avg=\frac{ 2^3-1^3 }{ 2-1 }=7\]
again 3<7 true

Answer 2

Yours is strictly increasing, not concave up.

Answer 3

Hopefully, you can see by the graph and the two examples that the slope of f(a) will always be less than the slope of a line from a point at x=a to a point at x=b (where a<b)