Question

The value of 'a' for which the function $f\left(x\right)=\{\begin{array}{ccc}-x^3+\cos a& 0<x<1& \\ x^2& x\ge 1& \end{array}$ has a local minimum at x = 1, is

• A. $-1$
• B. $1$
• C. $0$
• D. $-\frac{1}{2}$

Answer 1

Answer: (A), (D)

The correct options are
A $-1$
D $-\frac{1}{2}$

If $x=1$ is a point of minimum

$f\left(x\right)=\{\begin{array}{c}-x^3+\cos a;0<x<1\\ x^2;x\ge 1\end{array}$

$f(1+h)>f\left(1\right)$ and $f(1-h)>f\left(1\right)$

$-1+\cos a<-1$

$\cos a<0$

Therefore, $a=-1$ or $a=-\frac{1}{2}$ Answer.