1.

If x is real then find the minimum value of (x + 5)(x + 7)

Answer» Correct Answer - Option 2 : -1

Concept:

For a function y = f(x):

  • Relative (local) maxima are the points where the function f(x) changes its direction from increasing to decreasing.
  • Relative (local) minima are the points where the function f(x) changes its direction from decreasing to increasing.
  • At the points of relative (local) maxima or minima, f'(x) = 0.
  • At the points of relative (local) maxima, f''(x) < 0.
  • At the points of relative (local) minima, f''(x) > 0.

Calculation:

y = (x + 5)(x + 7)

⇒ y = x2 + 5x + 7x + 35

⇒ y = x2 + 12x + 35      ----(i)

By differentiating equation (i),

dy/dx = 0

⇒ 2x + 12 = 0      ----(ii)

⇒ x = - 6

Now by double differentiating equation (ii),

d2y/dx2 = 2 > 0

that means, the minimum value of equation (i) is at x = - 6

Minimum value of equation (i)

⇒ y = 36 - 72 + 35

⇒ y = - 1

∴ The minimum value is - 1.


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