if f(x) = x 2 /e x , then f′(–1) is equal to(a) -3e(b) 1/e(c) e(d) none of these© The Institute of Chartered Accountants of India

To find f'(x), we need to differentiate the function f(x) = x^2 / e^x using the quotient rule.

The quotient rule states that if we have a function of the form f(x) = g(x) / h(x), then the derivative f'(x) is given by (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2.

Applying the quotient rule to f(x) = x^2 / e^x, we have: g(x) = x^2 h(x) = e^x

Taking the derivatives of g(x) and h(x), we get: g'(x) = 2x h'(x) = e^x

Now, we can substitute these values into the quotient rule formula: f'(x) = (2x * e^x - x^2 * e^x) / (e^x)^2

Simplifying further, we have: f'(x) = (2x * e^x - x^2 * e^x) / e^(2x)

To find f'(-1), we substitute x = -1 into the derivative formula: f'(-1) = (2(-1) * e^(-1) - (-1)^2 * e^(-1)) / e^(-2)

Simplifying this expression, we get: f'(-1) = (-2e^(-1) + e^(-1)) / e^(-2) f'(-1) = (-e^(-1)) / e^(-2) + e^(-1) / e^(-2) f'(-1) = -e^(2) + e^(1)

Therefore, f'(-1) is equal to -e^(2) + e^(1).

None of the given options (a) -3e, (b) 1/e, (c) e, (d) none of these) match the value of f'(-1).