Find the maximum and the minimum values, if any, without using derivat

1.	Find the maximum and the minimum values, if any, without using derivatives of the following functions : f(x) = 16x2 –16x + 28 on R
Answer» We have, f(x) = 16x² – 16x + 28 on R = 16x² – 16x + 4 + 24 = (4x – 2)² + 24 Now, (4x – 2)² ≥ 0 for all x ∈ R = (4x – 2)² + 24 ≥ 24 for all x ∈ R = f(x) ≥ f (\(\frac{1}{2}\)) Thus, the minimum value of f(x) is 24 at x = (\(\frac{1}{2}\)) Hence, f(x) can be made large as possibly by giving difference value to x. Thus, maximum values does not exist.

Find the maximum and the minimum values, if any, without using derivatives of the following functions : f(x) = 16x2 –16x + 28 on R

Answer»

We have,

f(x) = 16x² – 16x + 28 on R

= 16x² – 16x + 4 + 24

= (4x – 2)² + 24

Now,

(4x – 2)² ≥ 0 for all x ∈ R

= (4x – 2)² + 24 ≥ 24 for all x ∈ R

= f(x) ≥ f (\(\frac{1}{2}\))

Thus, the minimum value of f(x) is 24 at x = (\(\frac{1}{2}\))

Hence,

f(x) can be made large as possibly by giving difference value to x.

Thus, maximum values does not exist.