Question

Find the local maxima of the function $f\left(x\right)=-x^2+7x-12$

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Answer 1

We saw that A differentiable function f(x) will have a local maximum at x = c, if f’(c) = 0 , f’(c-h) > 0 & f’(c+h) < 0 .

So to find the local maximum at x = c, we will differentiate f(x) and equate to zero.

f’(x) = 0

f’(x) = -2x + 7 = 0

Or x = 3.5

Now we have to check whether the sufficient condition is also satisfied.

f’(3.5 - h) = -2 (3.5 - h) + 7

= 2h

Which is positive as we know “h” is infinitesimal positive number.

f’(3.5 + h) = -2 (3.5 + h) + 7

= -2h

Which is negative.

As f’(x) is changing its sign from negative to positive we can say there is a maximum for f(x) at x = 3.5.